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

Goal state before:
pp: case hX
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable f (map X P)

pp: case hY
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable g (map X P)

pp: case hZ
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEMeasurable X P

Tactic:
· rw [hX.map_eq]
    exact hf.aestronglyMeasurable

Goal state after:
case hY
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEStronglyMeasurable g (map X P)

case hZ
Ω : Type u_1
𝓧 : Type u_2
mΩ : MeasurableSpace Ω
m𝓧 : MeasurableSpace 𝓧
X : Ω → 𝓧
μ : Measure 𝓧
P : Measure Ω
hX : HasLaw X μ P
f g : 𝓧 → ℝ
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ AEMeasurable X P
Module: Mathlib.Probability.Moments.Variance

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
mE : MeasurableSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
mF : MeasurableSpace F
μ : Measure E
inst✝¹ : IsProbabilityMeasure μ
ν : Measure F
inst✝ : IsProbabilityMeasure ν
L : StrongDual ℝ (E × F)
hLμ : MemLp (⇑(L ∘SL ContinuousLinearMap.inl ℝ E F)) 2 μ
hLν : MemLp (⇑(L ∘SL ContinuousLinearMap.inr ℝ E F)) 2 ν
⊢ Var[⇑L; μ.prod ν] = Var[⇑(L ∘SL ContinuousLinearMap.inl ℝ E F); μ] + Var[⇑(L ∘SL ContinuousLinearMap.inr ℝ E F); ν]

Tactic:
by
  have : L = fun x : E × F ↦ L.comp (.inl ℝ E F) x.1 + L.comp (.inr ℝ E F) x.2 := by
    ext; rw [L.comp_inl_add_comp_inr]
  rw [this, variance_add_prod hLμ hLν]

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

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
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)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
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'
hYt : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
⊢ (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i) →
    (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i) → (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, (κ a)[f i | Y i ⁻¹' t i]

Tactic:
hYt

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
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)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
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'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
⊢ (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i) → (κ a)[⋂ i ∈ s, f i | ⋂ i, Y i ⁻¹' t i] = ∏ i ∈ s, (κ a)[f i | Y i ⁻¹' t i]
Module: Mathlib.Probability.Independence.Kernel.IndepFun

Goal state before:
pp: α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
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)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
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'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
⊢ ((κ a) (⋂ i, Y i ⁻¹' t i))⁻¹ * (κ a) (⋂ i, g i) = (∏ i, (κ a) (Y i ⁻¹' t i))⁻¹ * (κ a) (⋂ i, g i)

Tactic:
hYt

Goal state after:
α : Type u_1
Ω : Type u_2
ι : Type u_3
β : Type u_4
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
mβ : MeasurableSpace β
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)) κ μ
hf : ∀ i ∈ s, MeasurableSet (f i)
hy✝ : ∀ᵐ (a : α) ∂μ, ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
ht : ∀ (i : ι), MeasurableSet (t i)
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'
hYt✝ : ∀ (i : ι), MeasurableSet (Y i ⁻¹' t i)
hg✝ : ∀ (i : ι), MeasurableSet (g i)
a : α
hy : ∀ i ∉ s, (κ a) (Y i ⁻¹' t i) ≠ 0
a✝ : IsProbabilityMeasure (κ a)
hYt : (κ a) (⋂ i, Y i ⁻¹' t i) = ∏ i, (κ a) (Y i ⁻¹' t i)
hg : (κ a) (⋂ i, g i) = ∏ i, (κ a) (g i)
⊢ (∏ i, (κ a) (Y i ⁻¹' t i))⁻¹ * (κ a) (⋂ i, g i) = (∏ i, (κ a) (Y i ⁻¹' t i))⁻¹ * (κ a) (⋂ i, g i)
Module: Mathlib.MeasureTheory.Function.Intersectivity

Goal state before:
pp: case refine_3.refine_2
α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u : Finset ℕ), ((⋂ n ∈ u, s n) \ N).Nonempty → 0 < μ (⋂ n ∈ u, s n)
f : ℕ → α → ℝ≥0∞ := fun n ↦ (↑(n + 1))⁻¹ • ∑ k ∈ Finset.range (n + 1), (s k).indicator 1
hfapp : ∀ (n : ℕ) (a : α), f n a = (↑(n + 1))⁻¹ * ∑ k ∈ Finset.range (n + 1), (s k).indicator 1 a
hf : ∀ (n : ℕ), Measurable (f n)
hf₁ : ∀ (n : ℕ), f n ≤ 1
hrf : ∀ (n : ℕ), r ≤ ∫⁻ (a : α), f n a ∂μ
hμ : μ ≠ 0
this : ∫⁻ (x : α), limsup (fun x_1 ↦ f x_1 x) atTop ∂μ ≤ μ univ
x : α
hxN : x ∉ N
hx : r / μ univ ≤ limsup (fun x_1 ↦ f x_1 x) atTop
u : Set ℕ
hux : u ⊆ {n | x ∈ s n}
hu : u.Finite
⊢ 0 < μ (⋂ n ∈ u, s n)

Tactic:
· simp_rw [← hu.mem_toFinset]
      exact hN₁ _ ⟨x, mem_iInter₂.2 fun n hn ↦ hux <| hu.mem_toFinset.1 hn, hxN⟩

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

Goal state before:
pp: case refine_2
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
int_f_ne_zero : ∀ (μ₀ : Measure G) [μ₀.IsHaarMeasure], ∫ (x : G), f x ∂μ₀ ≠ 0
⊢ HasCompactSupport fun x ↦ f (x * g)

Tactic:
have j : (fun x ↦ f (x * g)) = (f ∘ (Homeomorph.mulRight g)) := rfl

Goal state after:
case refine_2
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
int_f_ne_zero : ∀ (μ₀ : Measure G) [μ₀.IsHaarMeasure], ∫ (x : G), f x ∂μ₀ ≠ 0
j : (fun x ↦ f (x * g)) = f ∘ ⇑(Homeomorph.mulRight g)
⊢ HasCompactSupport fun x ↦ f (x * g)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
δ : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : Zero δ
f : γ →C_c δ
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : β
hx : x ∈ tsupport (⇑f ∘ ⇑g)
⊢ x ∈ ⇑g ⁻¹' tsupport f.toFun

Tactic:
tsupport,

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
δ : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : Zero δ
f : γ →C_c δ
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : β
hx : x ∈ tsupport (⇑f ∘ ⇑g)
⊢ x ∈ ⇑g ⁻¹' closure (Function.support f.toFun)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
δ : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : Zero δ
f : γ →C_c δ
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : β
hx : x ∈ tsupport (⇑f ∘ ⇑g)
o : Set γ
ho : IsOpen o
hgxo : g x ∈ o
⊢ (o ∩ Function.support f.toFun).Nonempty

Tactic:
tsupport,

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
δ : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : Zero δ
f : γ →C_c δ
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
x : β
hx : x ∈ closure (Function.support (⇑f ∘ ⇑g))
o : Set γ
ho : IsOpen o
hgxo : g x ∈ o
⊢ (o ∩ Function.support f.toFun).Nonempty
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ closure (Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun) ⊆ tsupport f₁.toFun ∪ tsupport f₂.toFun

Tactic:
tsupport,

Goal state after:
case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ closure (Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun) ⊆
    closure (Function.support f₁.toFun) ∪ tsupport f₂.toFun
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ closure (Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun) ⊆
    closure (Function.support f₁.toFun) ∪ tsupport f₂.toFun

Tactic:
tsupport,

Goal state after:
case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ closure (Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun) ⊆
    closure (Function.support f₁.toFun) ∪ closure (Function.support f₂.toFun)
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
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ

Tactic:
have heq : ∫ x in s, f x * g x ∂μ = ∫ x in s, f x ∂ν := by
    calc
      _ = ∫ x in s, (ρ x).toReal * f x ∂μ := by
        apply MeasureTheory.integral_congr_ae
        apply h_toReal_ae.mono
        intro x hx
        simp [hx, mul_comm]
      _ = _ := by
        have h := setIntegral_withDensity_eq_setIntegral_toReal_smul₀ hρ_ae hρ_top f hs_meas
        simp [ν, h]

Goal state after:
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
⊢ ∃ c ∈ s, ∫ (x : α) in s, f x * g x ∂μ = f c * ∫ (x : α) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: case hg
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
γ : Path a b
inst✝ : NormedSpace ℝ E
s : Set E
hω : ContinuousOn ω s
hγ : ContDiffOn ℝ 1 ⇑γ.extend I
hγs : ∀ (t : ↑I), γ t ∈ s
⊢ ContinuousOn (derivWithin ⇑γ.extend I) (Icc 0 1)

Tactic:
· exact hγ.continuousOn_derivWithin uniqueDiffOn_Icc_zero_one le_rfl

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
h : CurveIntegrable ω γ
⊢ CurveIntegrable (c • ω) γ

Tactic:
simpa [CurveIntegrable] using IntervalIntegrable.smul h c

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
H₂ : torusMap c R = fun θ x ↦ circleMap (c 0) (R 0) (θ 0)
⊢ ∫ (x : ℝ) in Icc 0 (2 * π),
      (∏ i, ↑(R i) * cexp (↑((MeasurableEquiv.funUnique (Fin 1) ℝ).symm x i) * I) * I) •
        f (torusMap c R ((MeasurableEquiv.funUnique (Fin 1) ℝ).symm x)) =
    ∫ (x : ℝ) in Icc 0 (2 * π), deriv (circleMap (c 0) (R 0)) x • f fun x_1 ↦ circleMap (c 0) (R 0) x ∂volume

Tactic:
H₂

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x ↦ 2 * π) = Icc 0 (2 * π)
H₂ : torusMap c R = fun θ x ↦ circleMap (c 0) (R 0) (θ 0)
⊢ ∫ (x : ℝ) in Icc 0 (2 * π),
      (∏ i, ↑(R i) * cexp (↑((MeasurableEquiv.funUnique (Fin 1) ℝ).symm x i) * I) * I) •
        f ((fun θ x ↦ circleMap (c 0) (R 0) (θ 0)) ((MeasurableEquiv.funUnique (Fin 1) ℝ).symm x)) =
    ∫ (x : ℝ) in Icc 0 (2 * π), deriv (circleMap (c 0) (R 0)) x • f fun x_1 ↦ circleMap (c 0) (R 0) x ∂volume
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
⊢ ⋃ i ∈ Finset.range N, Ioc (a + ↑i * ε) (a + ↑(i + 1) * ε) ⊆ ⋃ n, Ioc (y n - ε) (y n)

Tactic:
intro z
        simp only [Finset.mem_range, mem_iUnion, mem_Ioc, forall_exists_index, and_imp, y]
        refine fun n hn _ _ ↦ ⟨⟨n, hn⟩, ⟨by linarith, by simp_all [mul_comm ε _]⟩⟩

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

Goal state before:
pp: case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ x ∉ {a | (y n - ε < f a ∧ f a ≤ y n) ∧ a ∈ tsupport ⇑f}

Tactic:
simp_rw [mem_ofPred_eq, not_and_or, not_lt, not_le, or_assoc]

Goal state after:
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ
a ε : ℝ
hε : 0 < ε
N : ℕ
hf : range ⇑f ⊆ Ioo a (a + ↑N * ε)
b : ℝ := a + ↑N * ε
y : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)
hy : ∀ {n m : Fin N}, n < m → y n + ε ≤ y m
E : Fin N → Set X := fun n ↦ ⇑f ⁻¹' Ioc (y n - ε) (y n) ∩ tsupport ⇑f
m : Fin N
a✝¹ : m ∈ univ
n : Fin N
a✝ : n ∈ univ
hmn : m ≠ n
x : X
hx : y m - ε < f x ∧ f x ≤ y m ∧ x ∈ tsupport ⇑f
⊢ f x ≤ y n - ε ∨ y n < f x ∨ x ∉ tsupport ⇑f
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ (fun t ↦ charFun (Measure.map X P) t - taylorWithinEval (charFun (Measure.map X P)) 2 univ 0 t) =o[𝓝 0] fun t ↦ t ^ 2

Tactic:
convert! taylor_isLittleO_univ ?_

Goal state after:
case e'_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
x✝ : ℝ
⊢ x✝ = x✝ - 0

case convert_7
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
⊢ ContDiff ℝ (↑2) (charFun (Measure.map X P))
Module: Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic

Goal state before:
pp: E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ charFunDual (Measure.map (fun x ↦ x + r) μ) L = charFunDual μ L * cexp (↑(L r) * I)

Tactic:
rw [charFunDual_apply, charFunDual_apply, integral_map (by fun_prop) (by fun_prop),
    ← integral_mul_const]

Goal state after:
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
mE : MeasurableSpace E
μ : Measure E
inst✝ : BorelSpace E
r : E
L : StrongDual ℝ E
⊢ ∫ (x : E), cexp (↑(L (x + r)) * I) ∂μ = ∫ (a : E), cexp (↑(L a) * I) * cexp (↑(L r) * I) ∂μ
Module: Mathlib.Topology.Separation.CompletelyRegular

Goal state before:
pp: case refine_1
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
⊢ x ∈ f ⁻¹' Iio r

pp: case refine_2
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
⊢ f ⁻¹' Iio r ⊆ s

Tactic:
· simp [hf₀, hrclopen]

Goal state after:
case refine_2
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
⊢ f ⁻¹' Iio r ⊆ s
Module: Mathlib.MeasureTheory.Measure.FiniteMeasureProd

Goal state before:
pp: case refine_1
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}
a : Set α
b : Set β
ameas : MeasurableSet a
ha : μ.1 (frontier a) = 0
bmeas : MeasurableSet b
hb : μ.2 (frontier b) = 0
a' : Set α
b' : Set β
a'meas : MeasurableSet a'
ha' : μ.1 (frontier a') = 0
b'meas : MeasurableSet b'
hb' : μ.2 (frontier b') = 0
⊢ μ.1 (frontier (a ∩ a')) = 0

Tactic:
rw [null_iff_toMeasure_null] at ha ha' ⊢
      exact null_frontier_inter ha ha'

Goal state after:
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
⊢ Continuous fun x ↦ ∫ (a : A), (H.pullback f x) a ∂μA

Tactic:
Metric.continuous_iff'

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
⊢ ∀ (a : B),
    ∀ ε > 0, ∀ᶠ (x : B) in nhds a, dist (∫ (a : A), (H.pullback f x) a ∂μA) (∫ (a_2 : A), (H.pullback f a) a_2 ∂μA) < ε