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

Goal state before:
pp: G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : μ = μ'
⊢ μ U = (map (Quotient.mk (MulAction.orbitRel (↥Γ.op) G)) (ν.restrict s)) U

Tactic:
this

Goal state after:
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
s : Set G
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
V : Set (G ⧸ Γ)
meas_V : MeasurableSet V
neZeroV : μ V ≠ 0
hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ∞
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : μ = μ'
⊢ μ' U = (map (Quotient.mk (MulAction.orbitRel (↥Γ.op) G)) (ν.restrict s)) U
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case neg
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ QuotientMeasureEqMeasurePreimage ν μ

Tactic:
apply IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set (fund_dom_s := fund_dom_s)
    (meas_V := MeasurableSet.univ)

Goal state after:
case neg.neZeroV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ ≠ 0

case neg.hV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ = ν (QuotientGroup.mk ⁻¹' univ ∩ s)

case neg.neTopV
G : Type u_1
inst✝¹⁵ : Group G
inst✝¹⁴ : MeasurableSpace G
inst✝¹³ : TopologicalSpace G
inst✝¹² : IsTopologicalGroup G
inst✝¹¹ : BorelSpace G
inst✝¹⁰ : PolishSpace G
Γ : Subgroup G
inst✝⁹ : Γ.Normal
inst✝⁸ : T2Space (G ⧸ Γ)
inst✝⁷ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝⁶ : ν.IsMulLeftInvariant
inst✝⁵ : Countable ↥Γ
inst✝⁴ : ν.IsMulRightInvariant
inst✝³ : SigmaFinite ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : SigmaFinite μ
inst✝ : IsFiniteMeasure μ
hasFun : HasFundamentalDomain (↥Γ.op) G ν
s : Set G
h : ν s = μ univ
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
finiteCovol : μ univ < ∞
meas_s_ne_zero : ¬ν s = 0
⊢ μ univ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
ε : ℝ
x : AddCircle T
⊢ volume (Metric.closedBall x (2 * ε)) ≤ ↑(Real.toNNReal 2) * volume (Metric.closedBall x ε)

Tactic:
rw [volume_closedBall, volume_closedBall, ENNReal.ofNNReal_toNNReal 2,
    ← ENNReal.ofReal_mul zero_le_two]

Goal state after:
T : ℝ
hT : Fact (0 < T)
ε : ℝ
x : AddCircle T
⊢ ENNReal.ofReal (min T (2 * (2 * ε))) ≤ ENNReal.ofReal (2 * min T (2 * ε))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case e'_3.e_μ
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T))) = Measure.map (fun x ↦ ↑↑x) volume

Tactic:
[

Goal state after:
case e'_3.e_μ
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T))) = Measure.map (fun x ↦ ↑↑x) volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case e'_3.e_μ
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T))) = Measure.map (fun x ↦ ↑↑x) volume

Tactic:
this

Goal state after:
case e'_3.e_μ
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T))) =
    Measure.map (QuotientAddGroup.mk ∘ Subtype.val) volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume =
    ∫ (b : AddCircle T), f b ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T)))

Tactic:
[

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume =
    ∫ (b : AddCircle T), f b ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T)))
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume =
    ∫ (b : AddCircle T), f b ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T)))

Tactic:
this

Goal state after:
T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
t : ℝ
f : AddCircle T → E
m : MeasurableSet (Ioc t (t + T))
this✝ : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x
this : (fun x ↦ ↑↑x) = QuotientAddGroup.mk ∘ Subtype.val
⊢ ∫ (y : AddCircle T), f y ∂Measure.map (QuotientAddGroup.mk ∘ Subtype.val) volume =
    ∫ (y : AddCircle T), f y ∂Measure.map QuotientAddGroup.mk (volume.restrict (Ioc t (t + T)))
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ
⊢ AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (μ.restrict 𝓕))

Tactic:
[

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ
⊢ AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (μ.restrict 𝓕))
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ
⊢ ∫ (x : G), g (π x) * f x ∂μ = ∫ (x : G ⧸ Γ), automorphize (g ∘ π * f) x ∂Measure.map mk (μ.restrict 𝓕)

Tactic:
have H₂ : AEStronglyMeasurable (QuotientGroup.automorphize ((g ∘ π) * f)) μ_𝓕 := by
    simp_rw [H₀]
    exact hg.mul F_ae_measurable

Goal state after:
G : Type u_1
inst✝¹⁰ : Group G
inst✝⁹ : MeasurableSpace G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : BorelSpace G
μ : Measure G
Γ : Subgroup G
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ
inst✝⁵ : Countable ↥Γ
inst✝⁴ : MeasurableSpace (G ⧸ Γ)
inst✝³ : BorelSpace (G ⧸ Γ)
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ.IsMulRightInvariant
f : G → K
f_ℒ_1 : Integrable f μ
g : G ⧸ Γ → K
hg : AEStronglyMeasurable g (Measure.map mk (μ.restrict 𝓕))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ.restrict 𝓕)) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ.restrict 𝓕))
π : G → G ⧸ Γ := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ
H₂ : AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (μ.restrict 𝓕))
⊢ ∫ (x : G), g (π x) * f x ∂μ = ∫ (x : G ⧸ Γ), automorphize (g ∘ π * f) x ∂Measure.map mk (μ.restrict 𝓕)
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G' : Type u_1
inst✝¹⁰ : AddGroup G'
inst✝⁹ : MeasurableSpace G'
inst✝⁸ : TopologicalSpace G'
inst✝⁷ : IsTopologicalAddGroup G'
inst✝⁶ : BorelSpace G'
μ' : Measure G'
Γ' : AddSubgroup G'
𝓕' : Set G'
h𝓕 : IsAddFundamentalDomain (↥Γ'.op) 𝓕' μ'
inst✝⁵ : Countable ↥Γ'
inst✝⁴ : MeasurableSpace (G' ⧸ Γ')
inst✝³ : BorelSpace (G' ⧸ Γ')
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ'.IsAddRightInvariant
f : G' → K
f_ℒ_1 : Integrable f μ'
g : G' ⧸ Γ' → K
hg : AEStronglyMeasurable g (Measure.map mk (μ'.restrict 𝓕'))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ'.restrict 𝓕')) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ'.restrict 𝓕'))
π : G' → G' ⧸ Γ' := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ'
⊢ AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (μ'.restrict 𝓕'))

Tactic:
[

Goal state after:
G' : Type u_1
inst✝¹⁰ : AddGroup G'
inst✝⁹ : MeasurableSpace G'
inst✝⁸ : TopologicalSpace G'
inst✝⁷ : IsTopologicalAddGroup G'
inst✝⁶ : BorelSpace G'
μ' : Measure G'
Γ' : AddSubgroup G'
𝓕' : Set G'
h𝓕 : IsAddFundamentalDomain (↥Γ'.op) 𝓕' μ'
inst✝⁵ : Countable ↥Γ'
inst✝⁴ : MeasurableSpace (G' ⧸ Γ')
inst✝³ : BorelSpace (G' ⧸ Γ')
K : Type u_2
inst✝² : NormedField K
inst✝¹ : NormedSpace ℝ K
inst✝ : μ'.IsAddRightInvariant
f : G' → K
f_ℒ_1 : Integrable f μ'
g : G' ⧸ Γ' → K
hg : AEStronglyMeasurable g (Measure.map mk (μ'.restrict 𝓕'))
g_ℒ_infinity : essSup (fun x ↦ ‖g x‖ₑ) (Measure.map mk (μ'.restrict 𝓕')) ≠ ∞
F_ae_measurable : AEStronglyMeasurable (automorphize f) (Measure.map mk (μ'.restrict 𝓕'))
π : G' → G' ⧸ Γ' := mk
meas_π : Measurable π
H₀ : automorphize (g ∘ π * f) = g * automorphize f
H₁ : Integrable (g ∘ π * f) μ'
⊢ AEStronglyMeasurable (automorphize (g ∘ π * f)) (Measure.map mk (μ'.restrict 𝓕'))
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: c : ℂ
R : ℝ
⊢ (c +ᵥ R • range fun θ ↦ cexp (↑θ * I)) = sphere c |R|

Tactic:
rw [range_exp_mul_I, smul_sphere R 0 zero_le_one]
      simp

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

Goal state before:
pp: R : ℝ
z w : ℂ
hw : w ∈ ball z R
⊢ ∀ (θ : ℝ), circleMap z R θ - w ≠ 0

Tactic:
[

Goal state after:
R : ℝ
z w : ℂ
hw : w ∈ ball z R
⊢ ∀ (θ : ℝ), circleMap z R θ - w ≠ 0
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: R : ℝ
z w : ℂ
hw : w ∈ ball z R
this : ∀ (θ : ℝ), circleMap z R θ - w ≠ 0
⊢ Continuous fun θ ↦ (circleMap z R θ - w)⁻¹

Tactic:
exact Continuous.inv₀ (by fun_prop) this

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀) a₀

Tactic:
[

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Ioi a₀) μ
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀) a₀
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic

Goal state before:
pp: case negSucc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
t : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
this : ∀ (m : ℕ), ∫ (x : ℝ) in 0..m • T, f x = m • ∫ (x : ℝ) in 0..T, f x
n : ℕ
h₀ : Int.negSucc n • T + (n + 1) • T = 0
⊢ ∫ (x : ℝ) in Int.negSucc n • T..0, f (x + (n + 1) • T) = (n + 1) • ∫ (x : ℝ) in 0..T, f (x + (n + 1) • T)

Tactic:
[

Goal state after:
case negSucc
E : Type u_1
inst✝¹ : NormedAddCommGroup E
f : ℝ → E
T : ℝ
inst✝ : NormedSpace ℝ E
hf : Periodic f T
t : ℝ
h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable f volume t₁ t₂
this : ∀ (m : ℕ), ∫ (x : ℝ) in 0..m • T, f x = m • ∫ (x : ℝ) in 0..T, f x
n : ℕ
h₀ : Int.negSucc n • T + (n + 1) • T = 0
⊢ ∫ (x : ℝ) in Int.negSucc n • T..0, f (x + (n + 1) • T) = (n + 1) • ∫ (x : ℝ) in 0..T, f (x + (n + 1) • T)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀) a₀

Tactic:
[

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
a₀ : ℝ
hf : IntegrableOn f (Iio a₀) μ
⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀) a₀
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ici a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ici b, f x ∂μ) (Ici a₀)

Tactic:
[

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Ici a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Ici b, f x ∂μ) (Ici a₀)
Module: Mathlib.MeasureTheory.Integral.DominatedConvergence

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Iic a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Iic b, f x ∂μ) (Iic a₀)

Tactic:
[

Goal state after:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
μ : Measure ℝ
f : ℝ → E
inst✝ : NullSingletonClass μ
a₀ : ℝ
hf : IntegrableOn f (Iic a₀) μ
⊢ ContinuousOn (fun b ↦ ∫ (x : ℝ) in Iic b, f x ∂μ) (Iic a₀)
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ∞
⊢ HasFiniteIntegral (fun x ↦ ν.real (Prod.mk x ⁻¹' s)) μ

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : (μ.prod ν) s ≠ ∞
⊢ HasFiniteIntegral (fun x ↦ ν.real (Prod.mk x ⁻¹' s)) μ
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) ∧ ∫⁻ (a : α), ‖∫ (y : β), ‖f (a, y)‖ ∂ν‖ₑ ∂μ < ∞

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) ∧ ∫⁻ (a : α), ‖∫ (y : β), ‖f (a, y)‖ ∂ν‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Integral.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
    (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) ∧
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ∞

Tactic:
this

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ (∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞) →
    (∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ ↔
      ∫⁻ (a : α), ENNReal.ofReal (∫⁻ (a_1 : β), ‖f (a, a_1)‖ₑ ∂ν).toReal ∂μ < ∞)

α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
f : α × β → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : α), ∀ᵐ (y : β) ∂ν, 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (x : α), ∫⁻ (y : β), ‖f (x, y)‖ₑ ∂ν ∂μ < ∞ → ∀ᵐ (x : α) ∂μ, ∫⁻ (a : β), ‖f (x, a)‖ₑ ∂ν < ∞
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : |R| = R
⊢ ∀ z ∈ sphere c |R|, ‖f z‖ ≤ C

Tactic:
this

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : |R| = R
⊢ ∀ z ∈ sphere c R, ‖f z‖ ≤ C
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : |R| = R
⊢ 2 * π * |R| * C = 2 * π * R * C

Tactic:
this

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : |R| = R
⊢ 2 * π * R * C = 2 * π * R * C
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case e'_9
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
⊢ (z - w) ^ n = ↑(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (↑n + 1)

Tactic:
have hn' : (n + 1 : ℂ) ≠ 0 := by
        rwa [Ne, ← eq_neg_iff_add_eq_zero, ← Int.cast_one, ← Int.cast_neg, Int.cast_inj]

Goal state after:
case e'_9
n : ℤ
hn : n ≠ -1
c w : ℂ
R : ℝ
H : w ∈ sphere c |R| → -1 ≤ n
z : ℂ
hne : z ≠ w ∨ -1 ≤ n
hn' : ↑n + 1 ≠ 0
⊢ (z - w) ^ n = ↑(n + 1) * (id z - w) ^ (n + 1 - 1) * 1 / (↑n + 1)