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

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (μs i) F) L ≤ μ F
u : ℕ → ι
hu : Tendsto u atTop L
⊢ Tendsto (μs ∘ u) atTop (𝓝 μ)

Tactic:
apply tendsto_of_forall_isClosed_limsup_le_nat fun F hF ↦ le_trans ?_ (h F hF)

Goal state after:
Ω : Type u_1
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : ProbabilityMeasure Ω
μs : ι → ProbabilityMeasure Ω
L : Filter ι
inst✝ : L.IsCountablyGenerated
h : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (μs i) F) L ≤ μ F
u : ℕ → ι
hu : Tendsto u atTop L
F : Set Ω
hF : IsClosed[inst✝²] F
⊢ limsup (fun i ↦ ((μs ∘ u) i) F) atTop ≤ limsup (fun i ↦ (μs i) F) L
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
β : Type u_4
β' : Type u_5
γ : Type u_6
γ' : Type u_7
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
mγ : MeasurableSpace γ
mγ' : MeasurableSpace γ'
φ : β → γ
ψ : β' → γ'
hfg : IndepFun f g κ μ
hf : AEMeasurable f (⇑κ ∘ₘ μ)
hg : AEMeasurable g (⇑κ ∘ₘ μ)
hφ : AEMeasurable φ (Measure.map f (⇑κ ∘ₘ μ))
hψ : AEMeasurable ψ (Measure.map g (⇑κ ∘ₘ μ))
⊢ IndepFun (φ ∘ f) (ψ ∘ g) κ μ

Tactic:
by
  have h : IndepFun ((hφ.mk φ) ∘ f) ((hψ.mk ψ) ∘ g) κ μ := by
    refine IndepFun.comp hfg hφ.measurable_mk hψ.measurable_mk
  have hφ_ae := ae_of_ae_map hf hφ.ae_eq_mk
  have hψ_ae := ae_of_ae_map hg hψ.ae_eq_mk
  refine IndepFun.congr' h ?_ ?_
  · filter_upwards [Measure.ae_ae_of_ae_comp (hφ_ae)] with a haφ
    filter_upwards [haφ] with ω hωφ
    simp [hωφ]
  · filter_upwards [Measure.ae_ae_of_ae_comp (hψ_ae)] with a haψ
    filter_upwards [haψ] with ω hωψ
    simp [hωψ]

Goal state after:
Module: Mathlib.Probability.Independence.Basic

Goal state before:
pp: ι : Type u_6
Ω : Type u_7
α : Type u_8
β : Type u_9
mΩ : MeasurableSpace Ω
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure Ω
X : ι → Ω → α
Y : ι → Ω → β
f : ι → Set Ω
t : ι → Set β
s : Finset ι
inst✝ : Finite ι
hY : ∀ (i : ι), Measurable (Y i)
hindep : iIndepFun (fun i ω ↦ (X i ω, Y i ω)) μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy : ∀ i ∉ s, μ (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
this : IsProbabilityMeasure μ
val✝ : Fintype ι
g : ι → Set Ω := fun i' ↦ if i' ∈ s then Y i' ⁻¹' t i' ∩ f i' else Y i' ⁻¹' t i'
⊢ (μ (⋂ i, Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i) = (∏ i, μ (Y i ⁻¹' t i))⁻¹ * μ (⋂ i, g i)

Tactic:
by
      rw [hindep.meas_iInter]
      exact fun i ↦ ⟨.univ ×ˢ t i, MeasurableSet.univ.prod (ht _), by ext; simp⟩

Goal state after:
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: case refine_2.refine_2
α : Type u_1
Ω : Type u_2
β : Type u_4
γ : Type u_6
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
f : Ω → β
g : Ω → γ
hf : Measurable f
hg : Measurable g
h :
  ∀ {s : Set α} {t : Set β} {u : Set γ},
    MeasurableSet s →
      MeasurableSet t →
        MeasurableSet u → (μ ⊗ₘ κ.map fun ω ↦ (f ω, g ω)) (s ×ˢ t ×ˢ u) = (μ ⊗ₘ (κ.map f ×ₖ κ.map g)) (s ×ˢ t ×ˢ u)
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ Measurable fun a ↦ (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)

Tactic:
exact (Kernel.measurable_coe _ (hf hs)).mul (Kernel.measurable_coe _ (hg ht))

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

Goal state before:
pp: Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : SecondCountableTopology Ω
S : Set (Set Ω)
ν : ProbabilityMeasure Ω
h : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
G : Set Ω
hG : IsOpen[inst✝¹] G
r : ℝ≥0
hr : r < ν G
T : Set (Set Ω)
TS : T ⊆ S
T_count : T.Countable
hT : ⋃ t ∈ T, t = G
this✝ : T.Nonempty
f : ℕ → Set Ω
hf : T = range f
G_eq : G = ⋃ n, f n
this : Tendsto (fun i ↦ ν (accumulate f i)) atTop (𝓝 (ν G))
n : ℕ
hn : r < ν (accumulate f n)
⊢ ∃ T, (∀ t ∈ T, t ∈ S) ∧ r < ν (⋃ t ∈ T, t) ∧ ⋃ t ∈ T, t ⊆ G

Tactic:
refine ⟨(Finset.range (n + 1)).image f, by grind, ?_, ?_⟩

Goal state after:
case refine_1
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : SecondCountableTopology Ω
S : Set (Set Ω)
ν : ProbabilityMeasure Ω
h : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
G : Set Ω
hG : IsOpen[inst✝¹] G
r : ℝ≥0
hr : r < ν G
T : Set (Set Ω)
TS : T ⊆ S
T_count : T.Countable
hT : ⋃ t ∈ T, t = G
this✝ : T.Nonempty
f : ℕ → Set Ω
hf : T = range f
G_eq : G = ⋃ n, f n
this : Tendsto (fun i ↦ ν (accumulate f i)) atTop (𝓝 (ν G))
n : ℕ
hn : r < ν (accumulate f n)
⊢ r < ν (⋃ t ∈ Finset.image f (Finset.range (n + 1)), t)

case refine_2
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : SecondCountableTopology Ω
S : Set (Set Ω)
ν : ProbabilityMeasure Ω
h : ∀ (u : Set Ω), IsOpen[inst✝¹] u → ∀ x ∈ u, ∃ s ∈ S, s ∈ 𝓝 x ∧ s ⊆ u
G : Set Ω
hG : IsOpen[inst✝¹] G
r : ℝ≥0
hr : r < ν G
T : Set (Set Ω)
TS : T ⊆ S
T_count : T.Countable
hT : ⋃ t ∈ T, t = G
this✝ : T.Nonempty
f : ℕ → Set Ω
hf : T = range f
G_eq : G = ⋃ n, f n
this : Tendsto (fun i ↦ ν (accumulate f i)) atTop (𝓝 (ν G))
n : ℕ
hn : r < ν (accumulate f n)
⊢ ⋃ t ∈ Finset.image f (Finset.range (n + 1)), t ⊆ G
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ μ {ω | ↑c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} ≤ eVar[X; μ] / ↑c ^ 2

Tactic:
convert!
      meas_ge_le_mul_pow_eLpNorm_enorm μ two_ne_zero ENNReal.ofNat_ne_top (hX.sub B) A (by simp)
    using 1

Goal state after:
case e'_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ μ {ω | ↑c ≤ |X ω - ∫ (x : Ω), X x ∂μ|} = μ {x | ↑c ≤ ‖(X - fun x ↦ ∫ (x : Ω), X x ∂μ) x‖ₑ}

case e'_4
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEStronglyMeasurable X μ
c : ℝ≥0
hc : c ≠ 0
A : ↑c ≠ 0
B : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ
⊢ eVar[X; μ] / ↑c ^ 2 = (↑c)⁻¹ ^ ENNReal.toReal 2 * eLpNorm (X - fun x ↦ ∫ (x : Ω), X x ∂μ) 2 μ ^ ENNReal.toReal 2
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : SecondCountableTopology E
mE : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
X : Ω → E
hX : HasLaw X μ P
⊢ ∫ (x : Ω), (id ∘ X) x ∂P = ∫ (x : E), x ∂μ

Tactic:
hX.integral_comp aestronglyMeasurable_id

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : SecondCountableTopology E
mE : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
X : Ω → E
hX : HasLaw X μ P
⊢ ∫ (x : E), id x ∂μ = ∫ (x : E), x ∂μ
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ X' ⟂ᵢ[μ'] Y'

Tactic:
indepFun_iff_map_prod_eq_prod_map_map,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ Measure.map (fun ω ↦ (X' ω, Y' ω)) μ' = (Measure.map X' μ').prod (Measure.map Y' μ')

case hf
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable X' μ'

case hg
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
inst✝² : MeasurableSpace δ
μ : Measure γ
μ' : Measure δ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure μ'
X : γ → α
X' : δ → α
Y : γ → β
Y' : δ → β
h_indep : X ⟂ᵢ[μ] Y
h_ident : IdentDistrib (fun ω ↦ (X ω, Y ω)) (fun ω ↦ (X' ω, Y' ω)) μ μ'
⊢ AEMeasurable Y' μ'
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ A.indicator 1 ⟂ᵢ[P] X

pp: Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P

Tactic:
· exact hA1.indicator_indepFun 1 hA2

Goal state after:
Ω : Type u_5
𝓧 : Type u_6
m mΩ : MeasurableSpace Ω
P : Measure Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
A : Set Ω
E : Type u_7
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ mΩ
f : 𝓧 → E
hA1 : Indep m (MeasurableSpace.comap X m𝓧) P
hX : AEMeasurable X P
hA2 : MeasurableSet A
hf : AEStronglyMeasurable f (Measure.map X P)
⊢ AEMeasurable (A.indicator 1) P
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
u : ι
hu : u > i
⊢ Set.Ioo i u ∈ 𝓝[>] i

Tactic:
mem_nhdsWithin_iff_exists_mem_nhds_inter

Goal state after:
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
inst✝ : PartialOrder ι
𝓕 : Filtration ι m
this✝ : TopologicalSpace ι := Preorder.topology ι
this : OrderTopology ι
i : ι
hne : (𝓝[>] i).NeBot
u : ι
hu : u > i
⊢ ∃ u_1 ∈ 𝓝 i, u_1 ∩ Set.Ioi i ⊆ Set.Ioo i u
Module: Mathlib.Probability.Process.Filtration

Goal state before:
pp: ι : Type u_2
inst✝² : Preorder ι
X : ι → Type u_4
inst✝¹ : (i : ι) → MeasurableSpace (X i)
inst✝ : LocallyFiniteOrderBot ι
i : ι
⊢ ↑piLE i = MeasurableSpace.comap (frestrictLe i) MeasurableSpace.pi

Tactic:
by
  apply le_antisymm
  · simp_rw [piLE, ← piCongrLeft_comp_frestrictLe, ← MeasurableEquiv.coe_piCongrLeft, ← comap_comp]
    exact MeasurableSpace.comap_mono <| Measurable.comap_le (by fun_prop)
  · rw [← piCongrLeft_comp_restrictLe, ← MeasurableEquiv.coe_piCongrLeft, ← comap_comp]
    exact MeasurableSpace.comap_mono <| Measurable.comap_le (by fun_prop)

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

Goal state before:
pp: case neg.refine_1
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
⊢ ∀ ⦃s t : Set α⦄,
    s ⊆ t → (∀ (i : ι), eLpNorm (t.indicator (f i)) p μ ≤ ε) → ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ ε

Tactic:
intro s t hst ht i

Goal state after:
case neg.refine_1
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 (s✝ᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal
s t : Set α
hst : s ⊆ t
ht : ∀ (i : ι), eLpNorm (t.indicator (f i)) p μ ≤ ε
i : ι
⊢ eLpNorm (s.indicator (f i)) p μ ≤ ε
Module: Mathlib.MeasureTheory.Group.ModularCharacter

Goal state before:
pp: G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : LocallyCompactSpace G
g h : G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
mul_g_meas : Measurable fun x ↦ x * g
mul_h_meas : Measurable fun x ↦ x * h
ν : Measure G := haar
⊢ modularCharacterFun h * modularCharacterFun g =
    (map (fun x ↦ x * h) (map (fun x ↦ x * g) ν)).haarScalarFactor (map (fun x ↦ x * g) ν) * modularCharacterFun g

Tactic:
by
      rw [modularCharacterFun_eq_haarScalarFactor (map (· * g) ν) _]

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

Goal state before:
pp: case h
α : Type u_2
inst✝ : TopologicalSpace α
f g : α →C_c ℝ
h : α →C_c ℝ≥0
hh : (f + g).nnrealPart + h = f.nnrealPart + g.nnrealPart
x : α
hhx : max (f x + g x) 0 + ↑(h x) = max (f x) 0 + max (g x) 0
⊢ -(f x + g x) + max (f x + g x) 0 + ↑(h x) = -f x + max (f x) 0 + (-g x + max (g x) 0)

Tactic:
linear_combination hhx

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✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ [a -[ℝ] b], ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
t : ℝ
ht : t ∈ Ι 0 1
⊢ ‖(ω ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖

Tactic:
segment_eq_image_lineMap

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ ⇑(lineMap a b) '' Icc 0 1, ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
t : ℝ
ht : t ∈ Ι 0 1
⊢ ‖(ω ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
b : E
hb : b ∈ ball a δ
⊢ b ∈ s → ‖∫ᶜ (x : E) in Path.segment a b, ω x - (ω a) (b - a)‖ ≤ ε * ‖b - a‖

Tactic:
hbs

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
a : E
s : Set E
ω : E → E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ᶠ (x : E) in 𝓝[s] a, ContinuousWithinAt ω s x
ha : a ∈ s
ε : ℝ
hε : 0 < ε
δ : ℝ
hδ₀ : δ > 0
hδ : ball a δ ∩ s ⊆ {z | ContinuousWithinAt ω s z ∧ dist (ω z) (ω a) ≤ ε}
b : E
hb : b ∈ ball a δ
hbs : b ∈ s
⊢ ‖∫ᶜ (x : E) in Path.segment a b, ω x - (ω a) (b - a)‖ ≤ ε * ‖b - a‖
Module: Mathlib.MeasureTheory.MeasurableSpace.Invariants

Goal state before:
pp: case succ
α : Type u_1
inst✝ : MeasurableSpace α
f : α → α
n : ℕ
ihn : invariants f ≤ invariants f^[n]
⊢ invariants f ≤ invariants f^[n + 1]

Tactic:
| succ n ihn =>

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

Goal state before:
pp: case h1.h1.h2
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 : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
C : #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑{∅} ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
rw [mk_singleton]
    exact one_lt_aleph0.le.trans B

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

Goal state before:
pp: μ : Measure ℝ
inst✝ : IsFiniteMeasure μ
n : ℕ
hint : MemLp id (↑n) μ
t : ℝ
k : ℕ
hkn : k ∈ Finset.range (n + 1)
hint' : MemLp id (↑k) μ
⊢ (↑k !)⁻¹ * ↑t ^ k * iteratedDerivWithin k (charFun μ) univ 0 = (↑k !)⁻¹ * (↑t * I) ^ k * ↑(∫ (x : ℝ), x ^ k ∂μ)

Tactic:
simp [iteratedDeriv_charFun_zero hint', mul_pow, mul_comm, mul_assoc, mul_left_comm]

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

Goal state before:

Tactic:
_ ≤ |a| + a := neg_le_iff_add_nonneg'.mp <| neg_abs_le a
        _ ≤ |a| + a + ε' * (n + 1) := (le_add_iff_nonneg_right (|a| + a)).mpr <| Left.mul_nonneg
          (le_of_lt hε'.1) <| Left.add_nonneg (Nat.cast_nonneg' n) (zero_le_one' ℝ)
        _ ≤ _ := by rw [← add_assoc, le_add_iff_nonneg_right]; exact le_of_lt hε'.1

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

Goal state before:
pp: E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂μ ∗ ν = (∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂μ) * ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂ν

Tactic:
rw [integral_conv]

Goal state after:
E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ ∫ (x : E), ∫ (y : E), cexp (↑⟪x + y, t⟫ * I) ∂ν ∂μ =
    (∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂μ) * ∫ (x : E), cexp (↑⟪x, t⟫ * I) ∂ν

E : Type u_3
inst✝⁶ : MeasurableSpace E
μ ν : Measure E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : BorelSpace E
inst✝² : SecondCountableTopology E
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
t : E
⊢ Integrable (fun x ↦ cexp (↑⟪x, t⟫ * I)) (μ ∗ ν)
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
hX : #X < 𝔠
x : X
s : Set X
hxs : x ∈ s
hs : IsOpen[inst✝¹] s
f : X → ↑I
hfc : Continuous[inst✝¹, _] f
hf₀ : f x = 0
hf₁ : EqOn f 1 sᶜ
R : Set ↑I := range f
hR : #↑R < #↑I
r : ↑I
hr : r ∈ Rᶜ
hr' : ∀ (x : X), f x ≠ r
⊢ ∃ v ∈ {s | IsClopen s}, x ∈ v ∧ v ⊆ s

Tactic:
have hrclopen : f ⁻¹' Iio r = f ⁻¹' Iic r := by
    ext; simp [le_iff_lt_or_eq, hr']

Goal state after:
X : Type u
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
hX : #X < 𝔠
x : X
s : Set X
hxs : x ∈ s
hs : IsOpen[inst✝¹] s
f : X → ↑I
hfc : Continuous[inst✝¹, _] f
hf₀ : f x = 0
hf₁ : EqOn f 1 sᶜ
R : Set ↑I := range f
hR : #↑R < #↑I
r : ↑I
hr : r ∈ Rᶜ
hr' : ∀ (x : X), f x ≠ r
hrclopen : f ⁻¹' Iio r = f ⁻¹' Iic r
⊢ ∃ v ∈ {s | IsClopen s}, x ∈ v ∧ v ⊆ s
Module: Mathlib.MeasureTheory.Measure.DiracProba

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompletelyRegularSpace X
K : Set X
K_closed : IsClosed[inst✝¹] K
x : X
x_notin_K : x ∉ K
g : X → ↑unitInterval
g_cont : Continuous[inst✝¹, _] g
gx_zero : g x = 0
g_one_on_K : EqOn g 1 K
g_bdd : ∀ (x y : X), dist (↑(g x)).toNNReal (↑(g y)).toNNReal ≤ 1
g' : X →ᵇ ℝ≥0 := mkOfBound { toFun := fun x ↦ (↑(g x)).toNNReal, continuous_toFun := ⋯ } 1 g_bdd
f : X →ᵇ ℝ≥0 := 1 - g'
y : X
y_in_K : y ∈ K
⊢ f y = 0

Tactic:
simp [f, g', g_one_on_K y_in_K, tsub_self]

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
⊢ ∫ (v : E), cexp (↑(L v) * I) ∂μ ∗ ν = (∫ (v : E), cexp (↑(L v) * I) ∂μ) * ∫ (v : E), cexp (↑(L v) * I) ∂ν

Tactic:
rw [integral_conv]

Goal state after:
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
⊢ ∫ (x : E), ∫ (y : E), cexp (↑(L (x + y)) * I) ∂ν ∂μ =
    (∫ (v : E), cexp (↑(L v) * I) ∂μ) * ∫ (v : E), cexp (↑(L v) * I) ∂ν

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
⊢ Integrable (fun v ↦ cexp (↑(L v) * I)) (μ ∗ ν)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls

Goal state before:
pp: case e'_3
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
⊢ ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p✝) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p✝ + 1)) =
    ENNReal.ofReal ((∫ (x : F), Real.exp (-‖x‖ ^ p✝) ∂ν) / Real.Gamma (↑(finrank ℝ F) / p✝ + 1))

Tactic:
let ψ := @Homeomorph.toMeasurableEquiv E F tE mE _ _ _ _
      (@ContinuousLinearEquiv.toHomeomorph ℝ ℝ _ _ _ _ _ _ E tE _ F _ _ _ _ φ)
    -- The map `ψ` is measure preserving by construction
    have : @MeasurePreserving E F mE _ ψ μ ν :=
      @Measurable.measurePreserving E F mE _ ψ (@MeasurableEquiv.measurable E F mE _ ψ) _
    rw [← this.integral_comp']
    rfl

Goal state after:
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))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
⊢ ∃ x ∈ {μ | μ.mass ≤ C}, ↑f ≤ 𝓝 x

Tactic:
let μlim := RealRMK.rieszMeasure Λ'

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))
Λ' : (E →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := Λ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }
μlim : Measure E := RealRMK.rieszMeasure Λ'
⊢ ∃ x ∈ {μ | μ.mass ≤ C}, ↑f ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
S : Set (Measure E)
inst✝¹ : NormedAddCommGroup E
inst✝ : ProperSpace E
h : Tendsto (fun r ↦ ⨆ μ ∈ S, μ {x | r < ‖x‖}) atTop (𝓝 0)
⊢ IsTightMeasureSet S

Tactic:
by
  refine isTightMeasureSet_of_tendsto_measure_compl_closedBall (x := 0) ?_
  convert! h using 6 with r
  ext
  simp

Goal state after: