mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
β : Type u_3
β' : Type u_4
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
f : Ω → β
g : Ω → β'
hfm : Measurable f
hgm : Measurable g
⊢ f ⟂ᵢ[μ] g ↔
    ∀ {φ : β → ℝ} {ψ : β' → ℝ},
      Measurable φ →
        Measurable ψ →
          Integrable (φ ∘ f) μ →
            Integrable (ψ ∘ g) μ → integral μ (φ ∘ f * ψ ∘ g) = integral μ (φ ∘ f) * integral μ (ψ ∘ g)

Tactic:
by
  refine ⟨fun hfg _ _ hφ hψ _ _ => hfg.integral_comp_mul_comp
      hfm.aemeasurable hgm.aemeasurable hφ.aestronglyMeasurable hψ.aestronglyMeasurable, ?_⟩
  rw [IndepFun_iff]
  rintro h _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩
  specialize
    h (measurable_one.indicator hA) (measurable_one.indicator hB)
      ((integrable_const 1).indicator (hfm.comp measurable_id hA))
      ((integrable_const 1).indicator (hgm.comp measurable_id hB))
  rwa [← toReal_eq_toReal_iff' (measure_ne_top μ _), toReal_mul, ← measureReal_def,
    ← measureReal_def, ← measureReal_def, ← integral_indicator_one ((hfm hA).inter (hgm hB)),
    ← integral_indicator_one (hfm hA), ← integral_indicator_one (hgm hB), Set.inter_indicator_one]
  exact mul_ne_top (measure_ne_top μ _) (measure_ne_top μ _)

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInDistribution

Goal state before:
pp: case inr.inl
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
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : ∀ (x y : E), F x = F y
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))

Tactic:
specialize hF_lip x₀

Goal state after:
case inr.inl
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
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
hF_cont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] F
hF_lip : ∀ (y : E), F x₀ = F y
⊢ Tendsto (fun n ↦ ∫ (ω : E), F ω ∂Measure.map (Y n) μ'') l (𝓝 (∫ (ω : E), F ω ∂Measure.map Z μ'))
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : NormedAddCommGroup β
x✝ : MeasurableSpace α
f : ι → α → β
p : ℝ≥0∞
μ : Measure α
⊢ UnifTight f p μ ↔ ∀ ⦃ε : ℝ⦄, 0 < ε → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ENNReal.ofReal ε

Tactic:
refine ⟨fun hut rε hrε ↦ hut (Real.toNNReal_pos.mpr hrε), fun hut ε hε ↦ ?_⟩
  obtain ⟨s, hμs, hfε⟩ := hut hε
  use s, hμs; intro i
  exact (hfε i).trans_eq (ofReal_coe_nnreal (p := ε))

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: case neg.refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hε_top : ¬ε = ∞
s : Set α
hμs : μ s ≠ ∞
hfs : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal
⊢ sᶜ ∈ μ.cofinite

Tactic:
Measure.compl_mem_cofinite,

Goal state after:
case neg.refine_2
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hε_top : ¬ε = ∞
s : Set α
hμs : μ s ≠ ∞
hfs : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal
⊢ μ s < ∞
Module: Mathlib.MeasureTheory.Group.FoelnerFilter

Goal state before:
pp: G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g✝ h✝ : G
s : Set X
g h : G
i : ι
hi : μ (F i) ≠ 0
⊢ μ (g • s ∩ F i) / μ (F i) ≤ μ (h • s ∩ F i) / μ (F i) + μ ((g⁻¹ • F i) ∆ (h⁻¹ • F i)) / μ (F i)

Tactic:
rw [← tsub_le_iff_left, ← ENNReal.sub_div <| fun _ _ ↦ hi]

Goal state after:
G : Type u_1
X : Type u_2
inst✝³ : MeasurableSpace X
μ : Measure X
inst✝² : Group G
inst✝¹ : MulAction G X
ι : Type u_3
u : Ultrafilter ι
F : ι → Set X
inst✝ : SMulInvariantMeasure G X μ
hfoel : IsFoelner G μ (↑u) F
g✝ h✝ : G
s : Set X
g h : G
i : ι
hi : μ (F i) ≠ 0
⊢ (μ (g • s ∩ F i) - μ (h • s ∩ F i)) / μ (F i) ≤ μ ((g⁻¹ • F i) ∆ (h⁻¹ • F i)) / μ (F i)
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ (b ^ p⁻¹)⁻¹ * b ^ (-p⁻¹ * q) * (1 / p * Gamma ((q + 1) / p)) = b ^ (-(q + 1) / p) * (1 / p * Gamma ((q + 1) / p))

pp: p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ 0 < p

Tactic:
← rpow_neg_one,

Goal state after:
p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ (b ^ p⁻¹) ^ (-1) * b ^ (-p⁻¹ * q) * (1 / p * Gamma ((q + 1) / p)) = b ^ (-(q + 1) / p) * (1 / p * Gamma ((q + 1) / p))

p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ 0 < p
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic

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

Tactic:
apply le_trans _ (measure_mono h)

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
V : Set X
h : tsupport ⇑f ⊆ V
⊢ ↑(Λ f) ≤ (rieszMeasure Λ) (tsupport ⇑f)
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
r : ℝ
hr : 0 < r ∧ range f.toFun ⊆ Metric.ball 0 r
⊢ Metric.ball 0 r = Ioo (-r) r

Tactic:
simp [Real.ball_eq_Ioo]

Goal state after:
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : Ordinal.lift.{u, v} i ≤ (ℵ_ 1).ord
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
H : Cardinal.lift.{u, v} i.card ≤ ℵ_ 1
⊢ 2 ≤ Cardinal.lift.{v, u} (max (#↑s) 2)

Tactic:
rw [lift_max, Cardinal.lift_ofNat]

Goal state after:
α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : Ordinal.lift.{u, v} i ≤ (ℵ_ 1).ord
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
H : Cardinal.lift.{u, v} i.card ≤ ℵ_ 1
⊢ 2 ≤ max (Cardinal.lift.{v, u} #↑s) 2
Module: Mathlib.MeasureTheory.MeasurableSpace.Card

Goal state before:
pp: α : Type u
s : Set (Set α)
i✝¹ i✝ i : Ordinal.{v}
IH : ∀ y < i, y ≤ ω_ 1 → #↑(generateMeasurableRec s y) ≤ max (#↑s) 2 ^ ℵ₀
hi : Ordinal.lift.{u, v} i ≤ (ℵ_ 1).ord
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
H : Cardinal.lift.{u, v} i.card ≤ ℵ_ 1
⊢ 2 ≤ max (Cardinal.lift.{v, u} #↑s) 2

Tactic:
exact le_max_right _ _

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

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h : TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
x : X
K : Set X
hK : IsClosed[inst✝] K
hx✝ : x ∉ K
s : Set X
hs : s ∈ {s | IsClopen s}
hx : x ∈ s
hsK : s ⊆ Kᶜ
⊢ ∃ f, Continuous[inst✝, _] f ∧ f x = 0 ∧ EqOn f 1 K

Tactic:
refine ⟨sᶜ.indicator 1, ?_, by simpa, fun x hx ↦ indicator_of_mem ?_ _⟩

Goal state after:
case refine_1
X : Type u
inst✝ : TopologicalSpace X
h : TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
x : X
K : Set X
hK : IsClosed[inst✝] K
hx✝ : x ∉ K
s : Set X
hs : s ∈ {s | IsClopen s}
hx : x ∈ s
hsK : s ⊆ Kᶜ
⊢ Continuous[inst✝, _] (sᶜ.indicator 1)

case refine_2
X : Type u
inst✝ : TopologicalSpace X
h : TopologicalSpace.IsTopologicalBasis {s | IsClopen s}
x✝ : X
K : Set X
hK : IsClosed[inst✝] K
hx✝¹ : x✝ ∉ K
s : Set X
hs : s ∈ {s | IsClopen s}
hx✝ : x✝ ∈ s
hsK : s ⊆ Kᶜ
x : X
hx : x ∈ K
⊢ x ∈ sᶜ
Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: case pos
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoEMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ ν κ : Measure Ω
LPμν_finite : levyProkhorovEDist μ ν = ∞
⊢ levyProkhorovEDist μ κ ≤ levyProkhorovEDist μ ν + levyProkhorovEDist ν κ

Tactic:
simp [LPμν_finite]

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

Goal state before:
pp: E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
μ ν : Measure E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
L : StrongDual ℝ E
⊢ charFunDual (μ ∗ ν) L = charFunDual μ L * charFunDual ν L

Tactic:
simp_rw [charFunDual_apply]
  rw [integral_conv]
  · simp [add_mul, Complex.exp_add, integral_const_mul, integral_mul_const]
  · exact (integrable_const (1 : ℝ)).mono (by fun_prop) (by simp)

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

Goal state before:
pp: case h
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u

Tactic:
refine ⟨univ.pi (fun i ↦ ball (x i) (r i)), ⟨fun i ↦ ball (x i) (r i), rfl,
      fun i ↦ measurableSet_ball, fun i ↦ by simpa using hr i⟩, ?_, ?_⟩

Goal state after:
case h.refine_1
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ (univ.pi fun i ↦ ball (x i) (r i)) ∈ 𝓝 x

case h.refine_2
ι : Type u_1
α : ι → Type u_2
inst✝⁵ : Fintype ι
inst✝⁴ : (i : ι) → MeasurableSpace (α i)
inst✝³ : (i : ι) → TopologicalSpace (α i)
inst✝² : ∀ (i : ι), SecondCountableTopology (α i)
inst✝¹ : ∀ (i : ι), PseudoMetrizableSpace (α i)
inst✝ : ∀ (i : ι), OpensMeasurableSpace (α i)
μ : (i : ι) → ProbabilityMeasure (α i)
S : Set (Set ((i : ι) → α i)) :=
  {t | ∃ s, t = univ.pi s ∧ (∀ (i : ι), MeasurableSet (s i)) ∧ ∀ (i : ι), (μ i) (frontier (s i)) = 0}
this✝ : IsPiSystem S
this : (i : ι) → PseudoMetricSpace (α i) := fun i ↦ pseudoMetrizableSpacePseudoMetric (α i)
u : Set ((i : ι) → α i)
u_open : IsOpen[Pi.topologicalSpace] u
x : (i : ι) → α i
xu : x ∈ u
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ u
r : ι → ℝ
rpos : ∀ (i : ι), r i ∈ Ioo 0 ε
hr : ∀ (i : ι), ↑(μ i) (frontier (thickening (r i) {x i})) = 0
⊢ (univ.pi fun i ↦ ball (x i) (r i)) ⊆ u
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 : 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
⊢ y n ≤ a + ε' * ↑N

Tactic:
by simp_all [y, show (n : ℝ) + 1 ≤ N by norm_cast; lia]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.DistribChar

Goal state before:
pp: case e_a
G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ (DomMulAct.mk g' • DomMulAct.mk g • addHaar).addHaarScalarFactor (DomMulAct.mk g' • addHaar) =
    (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar

Tactic:
addHaarScalarFactor_domSMul

Goal state after:
case e_a
G : Type u_1
A : Type u_2
inst✝⁶ : Group G
inst✝⁵ : AddCommGroup A
inst✝⁴ : DistribMulAction G A
inst✝³ : TopologicalSpace A
inst✝² : IsTopologicalAddGroup A
inst✝¹ : LocallyCompactSpace A
inst✝ : ContinuousConstSMul G A
g✝ : G
this✝ : MeasurableSpace A := borel A
this : BorelSpace A
g g' : G
⊢ (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar = (DomMulAct.mk g • addHaar).addHaarScalarFactor addHaar
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
K : Set B
hK : IsCompact K
hf : ∀ x ∉ K, f x = 0
x : C
hx : x ∉ ⇑ψ '' K
this : ∀ (a : A), f (Function.invFun (⇑ψ) x * φ a) = 0
⊢ ∫ (a : A), (H.pullback f (Function.invFun (⇑ψ) x)) a ∂μA = 0

Tactic:
by simp [this, pullback_def]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Extension

Goal state before:
pp: case inr
A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
h :
  μC Set.univ * μA {1} <
    ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC)
hc : Set.InjOn (⇑ψ) U
b : B
key : (Function.support fun a ↦ f (b * φ a)).Subsingleton
a : A
ha : (Function.support fun a ↦ f (b * φ a)) = {a}
⊢ ∫ (x : A) in {a}, f (b * φ x) ∂μA ≤ (μA {1}).toReal

Tactic:
integral_singleton,

Goal state after:
case inr
A : Type u_1
B : Type u_2
C : Type u_3
inst✝¹⁷ : Group A
inst✝¹⁶ : Group B
inst✝¹⁵ : Group C
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : TopologicalSpace B
inst✝¹² : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst✝¹¹ : IsTopologicalGroup A
inst✝¹⁰ : IsTopologicalGroup B
inst✝⁹ : MeasurableSpace A
inst✝⁸ : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst✝⁷ : IsTopologicalGroup C
inst✝⁶ : LocallyCompactSpace B
inst✝⁵ : MeasurableSpace C
inst✝⁴ : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst✝³ : T2Space B
inst✝² : MeasurableSpace B
inst✝¹ : BorelSpace B
U : Set B
hU : IsOpen U
inst✝ : DiscreteTopology A
h✝ : μC Set.univ * μA {1} < (H.inducedMeasure μA μC) U
ho : 0 < μA {1}
ht : μA {1} < ⊤
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (⇑f) 1 K
hf2 : IsCompact (tsupport ⇑f)
hf3 : tsupport ⇑f ⊆ U
hf4 : ∀ (x : B), f x ∈ Set.Icc 0 1
h :
  μC Set.univ * μA {1} <
    ENNReal.ofReal (∫ (c : C), ((H.pushforward μA) { toContinuousMap := f, hasCompactSupport' := hf2 }) c ∂μC)
hc : Set.InjOn (⇑ψ) U
b : B
key : (Function.support fun a ↦ f (b * φ a)).Subsingleton
a : A
ha : (Function.support fun a ↦ f (b * φ a)) = {a}
⊢ μA.real {a} • f (b * φ a) ≤ (μA {1}).toReal
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 ((∫ (x : E) in {0}, Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))

Tactic:
integral_singleton,

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} = ENNReal.ofReal (μ.real {0} • Real.exp (-‖0‖ ^ p) / Real.Gamma (↑(finrank ℝ E) / p + 1))
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.real {0} • Real.exp (-‖0‖ ^ p) / Real.Gamma 1)

Tactic:
Real.Gamma_one,

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} = ENNReal.ofReal (μ.real {0} • Real.exp (-‖0‖ ^ p) / 1)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} = r • {x | ∑ i, |x i| ^ p < 1}

Tactic:
← Set.preimage_smul_inv₀ (ne_of_gt hr),

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p
h₂ : ∀ (x : ι → ℝ), 0 ≤ (∑ i, |x i| ^ p) ^ (1 / p)
hr : 0 < r
⊢ {x | (∑ i, |x i| ^ p) ^ (1 / p) < r} = (fun x ↦ r⁻¹ • x) ⁻¹' {x | ∑ i, |x i| ^ p < 1}
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_2
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = r • {x | ∑ i, ‖x i‖ ^ p < 1}

Tactic:
← Set.preimage_smul_inv₀ (ne_of_gt hr),

Goal state after:
case e'_2
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : ∀ (x : ι → ℂ), 0 ≤ ∑ i, ‖x i‖ ^ p
h₂ : ∀ (x : ι → ℂ), 0 ≤ (∑ i, ‖x i‖ ^ p) ^ (1 / p)
hr : 0 < r
⊢ {x | (∑ i, ‖x i‖ ^ p) ^ (1 / p) < r} = (fun x ↦ r⁻¹ • x) ⁻¹' {x | ∑ i, ‖x i‖ ^ p < 1}
Module: Mathlib.MeasureTheory.Measure.PreVariation

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
q : Subtype MeasurableSet
hq : q ∈ Q.parts
hq_eq : ↑q = ⋃ i, ↑q ∩ s i
t : ℕ → Subtype MeasurableSet := fun i ↦ ⟨↑q ∩ s i, ⋯⟩
ht_disj : Pairwise (Disjoint on Subtype.val ∘ t)
⊢ f ↑q ≤ ∑' (i : ℕ), f (↑q ⊓ ↑(s' i))

Tactic:
calc f q
            _ = f (⋃ i, q.val ∩ s i) := congrArg f hq_eq
            _ = f (⋃ i, (t i).val) := rfl
            _ ≤ ∑' i, f (t i) := hf t ht_disj
            _ = ∑' i, f (q ⊓ s' i) := rfl

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

Goal state before:
pp: X : Type u_1
inst✝ : MeasurableSpace X
f : Set X → ℝ≥0∞
hf : IsSigmaSubadditiveSetFun f
hf' : f ∅ = 0
s : ℕ → Set X
hs : ∀ (i : ℕ), MeasurableSet (s i)
hs' : Pairwise (Disjoint on s)
b : ℝ≥0∞
Q : Finpartition ⟨⋃ i, s i, ⋯⟩
hQ : b < ∑ p ∈ Q.parts, f ↑p
s' : ℕ → Subtype MeasurableSet := fun i ↦ ⟨s i, ⋯⟩
P : (i : ℕ) → Finpartition (s' i) := fun i ↦ Q.restrict ⋯
⊢ ∑' (i : ℕ), ∑ q ∈ Q.parts, f (↑q ⊓ ↑(s' i)) = ∑' (i : ℕ), ∑ p ∈ (P i).parts, f ↑p

Tactic:
congr 1; funext i
          exact (Q.sum_restrict _ (fun p => f p) hf').symm

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : MeasurableSpace X
μ : Measure X
K : Set X
hK : IsCompact K
hKsub : K ⊆ μ.supportᶜ
⊢ μ K = 0

Tactic:
refine hK.induction_on measure_empty ?_ ?_ ?_
  · exact fun _ _ hst ht ↦ measure_mono_null hst ht
  · exact fun _ _ hs ht ↦ measure_union_null hs ht
  · intro x hxK
    obtain ⟨U, hUnhds, hU0⟩ := notMem_support_iff_exists.1 (hKsub hxK)
    exact ⟨U, mem_nhdsWithin_of_mem_nhds hUnhds, hU0⟩

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

Goal state before:
pp: case hcd
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : HereditarilyLindelofSpace 𝕜
inst✝⁵ : CompleteSpace 𝕜
inst✝⁴ : MeasurableSpace 𝕜
inst✝³ : BorelSpace 𝕜
inst✝² : RCLike A
inst✝¹ : NormedAlgebra 𝕜 A
μ : Measure 𝕜
inst✝ : IsFiniteMeasure μ
a : A
ha : a ∉ ⇑(algebraMap 𝕜 A) '' μ.support
h : μ.support.Nonempty
this : 0 < infDist a (⇑(algebraMap 𝕜 A) '' μ.support)
s : Set A := ⋯
hs_z : s ∈ 𝓝 a
hs_μ : s ⊆ (⇑(algebraMap 𝕜 A) '' μ.support)ᶜ
resolvent_meas : ∀ᶠ (w : A) in 𝓝 a, AEStronglyMeasurable (resolvent w) μ
x : 𝕜
hx : x ∈ μ.support
w : A
hw : w ∈ s
⊢ ‖a - w‖ ≤ infDist a (⇑(algebraMap 𝕜 A) '' μ.support) / 2

Tactic:
· apply le_of_lt
          simpa [s, ← dist_eq_norm, dist_comm w a] using hw

Goal state after: