mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ∃ T, (∀ t ∈ T, t ∈ S) ∧ r < ν (⋃ t ∈ T, t) ∧ ⋃ t ∈ T, t ⊆ G

Tactic:
have : Tendsto (fun i ↦ ν (accumulate f i)) atTop (𝓝 (ν (⋃ i, f i))) :=
    (ENNReal.tendsto_toNNReal_iff (by simp) (by simp)).2 tendsto_measure_iUnion_accumulate

Goal state after:
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 (⋃ i, f i)))
⊢ ∃ T, (∀ t ∈ T, t ∈ S) ∧ r < ν (⋃ t ∈ T, t) ∧ ⋃ t ∈ T, t ⊆ G
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: case add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (a : Ω), f a * f' a ∂μ + ∫⁻ (a : Ω), f a * g a ∂μ =
    (∫⁻ (ω : Ω), f ω ∂μ) * (∫⁻ (a : Ω), f' a ∂μ + ∫⁻ (a : Ω), g a ∂μ)

Tactic:
left_distrib,

Goal state after:
case add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω), f ω * f' ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), f' ω ∂μ
h_ind_g' : ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ
h_measM_f' : Measurable f'
⊢ ∫⁻ (a : Ω), f a * f' a ∂μ + ∫⁻ (a : Ω), f a * g a ∂μ =
    (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (a : Ω), f' a ∂μ + (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (a : Ω), g a ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEMeasurable X μ
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (X ω - ∫ (x : Ω), X x ∂μ) ∂μ = Var[X; μ]

Tactic:
variance_eq_integral hX

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
hX : AEMeasurable X μ
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (X ω - ∫ (x : Ω), X x ∂μ) ∂μ = ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) ^ 2 ∂μ
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ Var[X; Measure.map Y μ] = Var[X ∘ Y; μ]

Tactic:
variance_eq_integral hX,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : Ω → ℝ
Ω' : Type u_3
mΩ' : MeasurableSpace Ω'
μ : Measure Ω'
Y : Ω' → Ω
hX : AEMeasurable X (Measure.map Y μ)
hY : AEMeasurable Y μ
⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂Measure.map Y μ) ^ 2 ∂Measure.map Y μ = Var[X ∘ Y; μ]
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ ∞ = ↑c * (∞ * ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ)

Tactic:
top_mul I,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ ∞ = ↑c * ∞
Module: Mathlib.Probability.Independence.Integration

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
⊢ False

Tactic:
have : ∞ < ∞ := calc
    ∞ = c * ((∫⁻ ω, ‖X ω‖ₑ ∂μ) * (∫⁻ ω, ‖Y ω‖ₑ ∂μ)) := by
      rw [H, top_mul I, mul_top (by simpa)]
    _ ≤ ∫⁻ ω, ‖B (X ω) (Y ω)‖ₑ ∂μ := by
      rw [← lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.enorm hY.enorm J,
        ← lintegral_const_mul'' _ (by fun_prop)]
      gcongr with ω
      simp [hB, ← mul_assoc]
    _ < ∞ := h'XY.2

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_5
F : Type u_6
G : Type u_7
inst✝⁸ : TopologicalSpace E
inst✝⁷ : ContinuousENorm E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : OpensMeasurableSpace E
inst✝⁴ : NormedAddGroup F
inst✝³ : MeasurableSpace F
inst✝² : OpensMeasurableSpace F
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousENorm G
X : Ω → E
Y : Ω → F
hXY : X ⟂ᵢ[μ] Y
B : E → F → G
c : ℝ≥0
hc : c ≠ 0
hB : ∀ (x : E) (y : F), ↑c * ‖x‖ₑ * ‖y‖ₑ ≤ ‖B x y‖ₑ
h'XY : Integrable (fun ω ↦ B (X ω) (Y ω)) μ
hX : AEStronglyMeasurable X μ
hY : AEStronglyMeasurable Y μ
h'Y : ¬Y =ᵐ[μ] 0
I : ∫⁻ (ω : Ω), ‖Y ω‖ₑ ∂μ ≠ 0
H : ∫⁻ (a : Ω), ‖X a‖ₑ ∂μ = ∞
J : (fun x ↦ ‖X x‖ₑ) ⟂ᵢ[μ] fun x ↦ ‖Y x‖ₑ
this : ∞ < ∞
⊢ False
Module: Mathlib.Probability.HasLaw

Goal state before:
pp: Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
P : Measure Ω
x : 𝓧
hX : X =ᵐ[P] fun x_1 ↦ x
⊢ map X P = P Set.univ • dirac x

Tactic:
map_congr hX

Goal state after:
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
P : Measure Ω
x : 𝓧
hX : X =ᵐ[P] fun x_1 ↦ x
⊢ map (fun x_1 ↦ x) P = P Set.univ • dirac x
Module: Mathlib.Probability.IdentDistrib

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
ν : Measure β
f : α → ℝ
g : β → ℝ
h : IdentDistrib f g μ ν
⊢ Var[f; μ] = Var[g; ν]

Tactic:
rw [variance, h.evariance_eq]; rfl

Goal state after:
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
hF_bounded : ∀ (x y : E), dist (F x) (F y) ≤ M
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''
⊢ ‖F (X n a)‖ ≤ ‖F x₀‖ + M

Tactic:
specialize hF_bounded (X n a) x₀

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.MeasureTheory.Function.Floor

Goal state before:
pp: R : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : FloorRing R
inst✝³ : TopologicalSpace R
inst✝² : OrderTopology R
inst✝¹ : MeasurableSpace R
inst✝ : OpensMeasurableSpace R
x : R
⊢ MeasurableSet (ceil ⁻¹' {⌈x⌉})

Tactic:
simpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.ContinuousCompMeasurePreserving

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : BorelSpace X
inst✝⁸ : R1Space X
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : BorelSpace Y
inst✝⁴ : R1Space Y
μ : Measure X
ν : Measure Y
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure ν
E : Type u_3
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
hp₀ : p ≠ 0
f : ↥(simpleFunc E p ν)
⊢ Continuous[_, PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] fun y ↦
    (compMeasurePreserving ⇑↑(↑f, y).2 ⋯) (↑f, y).1

Tactic:
induction f using Lp.simpleFunc.induction hp₀ hp with
  | add hfp hgp _ ihf ihg => exact ihf.add ihg
  | @indicatorConst c s hs hνs =>
    dsimp only [Lp.simpleFunc.coe_indicatorConst, Lp.indicatorConstLp_compMeasurePreserving]
    refine continuous_indicatorConstLp_set hp fun f ↦ ?_
    apply tendsto_measure_symmDiff_preimage_nhds_zero continuousAt_subtype_val _ f.2
      hs.nullMeasurableSet hνs.ne
    exact .of_forall Subtype.property

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

Goal state before:
pp: G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
μ : Measure G
inst✝ : μ.IsHaarMeasure
g : G
ν : Measure G := haar
⊢ modularCharacterFun g = (map (fun x ↦ x * g) μ).haarScalarFactor μ

Tactic:
obtain ⟨⟨f, f_cont⟩, f_comp, f_nonneg, f_one⟩ :
    ∃ f : C(G, ℝ), HasCompactSupport f ∧ 0 ≤ f ∧ f 1 ≠ 0 := exists_continuous_nonneg_pos 1

Goal state after:
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : LocallyCompactSpace G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
μ : Measure G
inst✝ : μ.IsHaarMeasure
g : G
ν : Measure G := haar
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }
f_nonneg : 0 ≤ { toFun := f, continuous_toFun := f_cont }
f_one : { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
⊢ modularCharacterFun g = (map (fun x ↦ x * g) μ).haarScalarFactor μ
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
⊢ (map (fun x ↦ x * h) (map (fun x ↦ x * g) ν)).haarScalarFactor (map (fun x ↦ x * g) ν) *
      (map (fun x ↦ x * g) ν).haarScalarFactor ν =
    (map (fun x ↦ x * (g * h)) ν).haarScalarFactor ν

Tactic:
simp only [map_map mul_h_meas mul_g_meas,
      comp_mul_right, ← haarScalarFactor_eq_mul]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n
hgg' : g =ᵐ[μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
⊢ g x = g' x → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x)) → Tendsto (fun n ↦ f' n x) atTop (𝓝 (g' x))

Tactic:
hgg'x

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n
hgg' : g =ᵐ[μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
hff'' : ∀ᵐ (x : α) ∂μ, ∀ (i : ℕ), f i x = f' i x
x : α
hff'x : ∀ (i : ℕ), f i x = f' i x
hgg'x : g x = g' x
⊢ Tendsto (fun n ↦ f n x) atTop (𝓝 (g x)) → Tendsto (fun n ↦ f' n x) atTop (𝓝 (g' x))
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-(x✝ ^ (1 / p)) ^ p)) =
    (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝))

Tactic:
← rpow_mul (le_of_lt hx) _ p,

Goal state after:
p q : ℝ
hp : 0 < p
hq : -1 < q
x✝ : ℝ
hx : x✝ ∈ Ioi 0
⊢ (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝ ^ (1 / p * p))) =
    (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝))
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
⊢ (ν s).toReal ≠ 0

Tactic:
simpa using! hν0

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✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
γ : Path a b
𝕝 : Type u_4
inst✝³ : RCLike 𝕝
inst✝² : NormedSpace 𝕝 F
inst✝¹ : NormedSpace 𝕝 E
inst✝ : LinearMap.CompatibleSMul E F 𝕝 𝕜
x✝ : ℝ
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ curveIntegralFun (fun t ↦ ContinuousLinearMap.restrictScalars 𝕝 (ω t)) γ x✝ = curveIntegralFun ω γ x✝

Tactic:
simp [curveIntegralFun_def]

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✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
γ : Path a b
𝕝 : Type u_4
inst✝² : RCLike 𝕝
inst✝¹ : NormedSpace 𝕝 F
inst✝ : SMulCommClass 𝕜 𝕝 F
c : 𝕝
⊢ curveIntegral (c • ω) γ = c • curveIntegral ω γ

Tactic:
let : NormedSpace ℝ F := .restrictScalars ℝ 𝕜 F
  simp [curveIntegral_def, intervalIntegral.integral_smul]

Goal state after:
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

pp: case mpr
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 (fun x ↦ ↑(f x) • g x) μ → AEMeasurable g (μ.withDensity fun x ↦ ↑(f x))

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:
case mpr
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 (fun x ↦ ↑(f x) • g x) μ → AEMeasurable g (μ.withDensity fun x ↦ ↑(f x))
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
⊢ (rieszContent (toNNRealLinear Λ)).measure ↑{ carrier := K, isCompact' := hK } ≤ ENNReal.ofReal (Λ f)

Tactic:
Content.measure_eq_content_of_regular _ (contentRegular_rieszContent (toNNRealLinear Λ))

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : MeasurableSpace X
inst✝¹ : BorelSpace X
Λ : (X →C_c ℝ) →ₚ[ℝ] ℝ
inst✝ : LocallyCompactSpace X
f : X →C_c ℝ
hf : ∀ (x : X), 0 ≤ f x
K : Set X
hK : IsCompact K
hfK : ∀ x ∈ K, f x = 1
⊢ (rieszContent (toNNRealLinear Λ)) { carrier := K, isCompact' := hK } ≤ ENNReal.ofReal (Λ f)
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 : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ #↑(⋃ j, ⋃ (_ : j < i), generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀

Tactic:
apply mk_biUnion_le_of_le_lift _ B _

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 : i ≤ ω_ 1
A : 𝔠 ≤ max (#↑s) 2 ^ ℵ₀
B : ℵ₀ ≤ max (#↑s) 2 ^ ℵ₀
⊢ ∀ j < i, #↑(generateMeasurableRec s j) ≤ max (#↑s) 2 ^ ℵ₀

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
⊢ Cardinal.lift.{u, v} i.card ≤ Cardinal.lift.{v, u} (max (#↑s) 2 ^ ℵ₀)
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: case h.refine_1
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
x : X
hx : x ∈ V₁ ⊓ V₂
⊢ f x < c

Tactic:
suffices ∀ x ∈ V₂.carrier, f x < c from this x (mem_of_mem_inter_right hx)

Goal state after:
case h.refine_1
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : T2Space X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
ε : ℝ
hε : 0 < ε
E : Set X
μ : Content X
hμ : μ.outerMeasure E ≠ ∞
hμ' : MeasurableSet E
c : ℝ
hfE : ∀ x ∈ E, f x < c
hε' : ε.toNNReal ≠ 0
V₁ : Opens X
hV₁ : E ⊆ ↑V₁ ∧ μ.outerMeasure ↑V₁ ≤ μ.outerMeasure E + ↑ε.toNNReal
V₂ : Opens X := { carrier := ⇑f ⁻¹' Iio c, is_open' := ⋯ }
x : X
hx : x ∈ V₁ ⊓ V₂
⊢ ∀ x ∈ V₂.carrier, f x < c
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : ∃ s V, (∀ (i : ↑↑s), V i ∈ 𝓝 x) ∧ K = ⋂ i, V i
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ

Tactic:
Finset.coe_sort_coe

Goal state after:
ι : Type u_1
X : Type u_2
t : ι → TopologicalSpace X
ht : ∀ (i : ι), CompletelyRegularSpace X
this : TopologicalSpace X := ⨅ i, t i
x : X
K : Set X
hK : IsOpen[⨅ i, t i] K
hxK : ∃ s V, (∀ (i : ↥s), V i ∈ 𝓝 x) ∧ K = ⋂ i, V i
⊢ ∃ f, Continuous[⨅ i, t i, _] f ∧ f x = 0 ∧ EqOn f 1 Kᶜ
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: 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
⊢ ball x.1 ε ×ˢ ball x.2 ε ⊆ u

Tactic:
by rwa [ball_prod_same]

Goal state after: