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

Goal state before:
pp: case inl.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ ↑f x ≤ ↑f y

Tactic:
]

Goal state after:
case inl.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ ↑f x ≤ ↑f y
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
g : X → ℝ
hf : ∀ x ∈ s, 0 ≤ f x
h : ∀ x ∈ s, f x ≤ g x
hg : IntegrableOn g s μ
h'f : ¬AEStronglyMeasurable f (μ.restrict s)
⊢ 0 ≤ ∫ (x : X) in s, g x ∂μ

Tactic:
]

Goal state after:
case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
g : X → ℝ
hf : ∀ x ∈ s, 0 ≤ f x
h : ∀ x ∈ s, f x ≤ g x
hg : IntegrableOn g s μ
h'f : ¬AEStronglyMeasurable f (μ.restrict s)
⊢ 0 ≤ ∫ (x : X) in s, g x ∂μ
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
this : 0 ≤ chaar K₀ K ∧ chaar K₀ K ≤ ↑(index ↑K ↑K₀)
⊢ 0 ≤ chaar K₀ K

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
this : 0 ≤ chaar K₀ K ∧ chaar K₀ K ≤ ↑(index ↑K ↑K₀)
⊢ 0 ≤ chaar K₀ K
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re (⟪x, y⟫ * ↑t) + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫ =
    normSq x * t * t + re (⟪x, y⟫ * ↑t) + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ↑f x - ↑g y = ↑g x - ↑g y
⊢ ↑f x = ↑g x

pp: case inr.hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ 0 ≤ ↑g x - ↑g y

pp: case inr.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ 0 ≤ ↑f x - ↑f y

Tactic:
]

Goal state after:
case inr
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ↑f x - ↑g y = ↑g x - ↑g y
⊢ ↑f x = ↑g x

case inr.hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ 0 ≤ ↑g x - ↑g y

case inr.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ 0 ≤ ↑f x - ↑f y
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ ↑g y ≤ ↑g x

Tactic:
]

Goal state after:
case inr.hp
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ ↑g y ≤ ↑g x
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' {1}

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {1})

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' {1}

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {1})
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫ =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝¹ : Countable ι
μ : Measure X
inst✝ : NormedAddCommGroup E
f : X → E
s : ι → Set X
hi : ∀ (i : ι), IntegrableOn f (s i) μ
h : Summable fun i ↦ ∫ (x : X) in s i, ‖f x‖ ∂μ
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
⊢ Summable fun a ↦ ↑(NNReal.mk (∫ (x : X) in s a, ↑‖f x‖₊ ∂μ) ⋯)

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝¹ : Countable ι
μ : Measure X
inst✝ : NormedAddCommGroup E
f : X → E
s : ι → Set X
hi : ∀ (i : ι), IntegrableOn f (s i) μ
h : Summable fun i ↦ ∫ (x : X) in s i, ‖f x‖ ∂μ
B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)
⊢ Summable fun a ↦ ↑(NNReal.mk (∫ (x : X) in s a, ↑‖f x‖₊ ∂μ) ⋯)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y =
    normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + normSq y

Tactic:
]

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

Goal state before:
pp: case inr.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ ↑f y ≤ ↑f x

Tactic:
]

Goal state after:
case inr.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ ↑f y ≤ ↑f x
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ U.Nonempty

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₀.toCompacts
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ U.Nonempty
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪y, x⟫ * t + re ⟪y, x⟫ * t + normSq y =
    normSq x * t * t + re ⟪y, x⟫ * t + re ⟪y, x⟫ * t + normSq y

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
t : ℝ
⊢ normSq x * t * t + re ⟪x, y⟫ * t + re ⟪x, y⟫ * t + normSq y = normSq x * t * t + 2 * re ⟪x, y⟫ * t + normSq y

Tactic:
by ring

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₂ - chaar K₀ K₁

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₂ - chaar K₀ K₁
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪⟪x, y⟫ • x - ⟪x, x⟫ • y, ⟪x, y⟫ • x - ⟪x, x⟫ • y⟫ = ↑(normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2))

Tactic:
]

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪⟪x, y⟫ • x - ⟪x, x⟫ • y, ⟪x, y⟫ • x - ⟪x, x⟫ • y⟫ = ↑(normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ
a b : R
hab : a ≤ b
⊢ c ≤ ↑(StieltjesFunction.const R c) a

pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ
a b : R
hab : a ≤ b
⊢ ContinuousWithinAt (↑(StieltjesFunction.const R c)) (Iic a) a

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ
a b : R
hab : a ≤ b
⊢ c ≤ ↑(StieltjesFunction.const R c) a

R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
c : ℝ
a b : R
hab : a ≤ b
⊢ ContinuousWithinAt (↑(StieltjesFunction.const R c)) (Iic a) a
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝⁵ : Countable ι
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace X
inst✝² : BorelSpace X
inst✝¹ : T2Space X
inst✝ : IsLocallyFiniteMeasure μ
f : C(X, E)
s : ι → Compacts X
hf : Summable fun i ↦ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)
i : ι
⊢ ‖∫ (x : X) in ↑(s i), ‖f x‖ ∂μ‖ ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝⁵ : Countable ι
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace X
inst✝² : BorelSpace X
inst✝¹ : T2Space X
inst✝ : IsLocallyFiniteMeasure μ
f : C(X, E)
s : ι → Compacts X
hf : Summable fun i ↦ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)
i : ι
⊢ ‖∫ (x : X) in ↑(s i), ‖f x‖ ∂μ‖ ≤ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
ι : Type u_5
inst✝⁵ : Countable ι
μ : Measure X
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace X
inst✝² : BorelSpace X
inst✝¹ : T2Space X
inst✝ : IsLocallyFiniteMeasure μ
f : C(X, E)
s : ι → Compacts X
hf : Summable fun i ↦ ‖ContinuousMap.restrict (↑(s i)) f‖ * μ.real ↑(s i)
⊢ IntegrableOn (⇑f) (⋃ i, ↑(s i)) μ

Tactic:
refine
    integrableOn_iUnion_of_summable_integral_norm
      (fun i => (map_continuous f).continuousOn.integrableOn_compact (s i).isCompact)
      (.of_nonneg_of_le (fun ι => integral_nonneg fun x => norm_nonneg _) (fun i => ?_) hf)
  rw [← (Real.norm_of_nonneg (integral_nonneg fun x => norm_nonneg _) : ‖_‖ = ∫ x in s i, ‖f x‖ ∂μ)]
  exact
    norm_setIntegral_le_of_norm_le_const (s i).isCompact.measure_lt_top
      fun x hx => (norm_norm (f x)).symm ▸ (f.restrict (s i : Set X)).norm_coe_le_norm ⟨x, hx⟩

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ U.Nonempty

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen[inst✝¹] U
h3U : 1 ∈ U
⊢ U.Nonempty
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ↑‖⟪x, y⟫‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 +
      ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =
    ↑(normSq x * (normSq x * normSq y) - normSq x * ‖⟪x, y⟫‖ ^ 2)

Tactic:
]

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ↑‖⟪x, y⟫‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 - ↑(normSq x) * ↑‖⟪x, y⟫‖ ^ 2 +
      ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =
    ↑(normSq x * (normSq x * normSq y) - normSq x * ‖⟪x, y⟫‖ ^ 2)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
f g : StieltjesFunction R
a b : R
h : a ≤ b
this : leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a
⊢ ofReal (↑f b + ↑g b - (leftLim (↑f) a + leftLim (↑g) a)) = ofReal (↑f b - leftLim (↑f) a + (↑g b - leftLim (↑g) a))

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
f g : StieltjesFunction R
a b : R
h : a ≤ b
this : leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a
⊢ ofReal (↑f b + ↑g b - (leftLim (↑f) a + leftLim (↑g) a)) = ofReal (↑f b - leftLim (↑f) a + (↑g b - leftLim (↑g) a))
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫

Tactic:
]

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
x : X
hx1 : ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(f + g) x
hx2 : ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x
hx3 : ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x
hx4 : ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x
hx5 : ↑↑(f + g) x = (↑↑f + ↑↑g) x
⊢ ↑↑f x + ↑↑g x = ↑↑f x + ↑↑g x

Tactic:
]

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂)

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : 2 * ‖⟪x, y⟫‖ * (2 * ‖⟪x, y⟫‖) - 4 * re ⟪x, x⟫ * re ⟪y, y⟫ ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫

Tactic:
]

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : 2 * ‖⟪x, y⟫‖ * (2 * ‖⟪x, y⟫‖) - 4 * re ⟪x, x⟫ * re ⟪y, y⟫ ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0

pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)

Tactic:
]

Goal state after:
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0

G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ (eLpNorm (↑↑f) p (μ.restrict s)).toReal ≤ (eLpNorm (↑↑f) p μ).toReal

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ (eLpNorm (↑↑f) p (μ.restrict s)).toReal ≤ (eLpNorm (↑↑f) p μ).toReal