mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.refine_2
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)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x

pp: 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)
⊢ NullMeasurableSet {x | f x ≤ g x} (μ.restrict s)

Tactic:
·

Goal state after:
case pos.refine_2
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)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ 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 ⊥
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
⊢ 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
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: 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)
⊢ NullMeasurableSet {x | f x ≤ g x} (μ.restrict s)

Tactic:
·

Goal state after:
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)
⊢ NullMeasurableSet {x | f x ≤ g x} (μ.restrict s)
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)
⊢ 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.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
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}
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)
⊢ 0 ≤ ↑f x - ↑f y

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)
⊢ 0 ≤ ↑f x - ↑f 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
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
⊢ IsClosed (eval ⁻¹' {1})
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
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
⊢ 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
⊢ 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
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
ha : 𝓝[<] a = ⊥
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a

pp: case inr
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
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a

Tactic:
·

Goal state after:
case inl
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
ha : 𝓝[<] a = ⊥
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
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
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a

Tactic:
·

Goal state after:
case inr
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
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) 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
⊢ 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
⊢ IsClosed (eval ⁻¹' Ici 0)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : 𝓝[<] a = ⊥
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a

pp: case inr
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a

Tactic:
·

Goal state after:
case inl
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : 𝓝[<] a = ⊥
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case pos
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
hzero : ⟪x, y⟫ = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y

pp: case neg
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
hzero : ¬⟪x, y⟫ = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y

Tactic:
·

Goal state after:
case pos
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
hzero : ⟪x, y⟫ = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a

Tactic:
·

Goal state after:
case inr
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≥0
f : StieltjesFunction R
a b : R
h : a ≤ b
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(c • f)) a = ↑c * leftLim (↑f) a
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case neg
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
hzero : ¬⟪x, y⟫ = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y

Tactic:
·

Goal state after:
case neg
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
hzero : ¬⟪x, y⟫ = 0
⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case e'_4.e'_5
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ normSq x * (t * t) = normSq (⟪x, y⟫ • x) * (t / ‖⟪x, y⟫‖) * (t / ‖⟪x, y⟫‖)

pp: case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫‖ * t = 2 * re ⟪⟪x, y⟫ • x, y⟫ * (t / ‖⟪x, y⟫‖)

Tactic:
·

Goal state after:
case e'_4.e'_5
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ normSq x * (t * t) = normSq (⟪x, y⟫ • x) * (t / ‖⟪x, y⟫‖) * (t / ‖⟪x, 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
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
⊢ 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
⊢ IsClosed (eval ⁻¹' Ici 0)
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫‖ * t = 2 * re ⟪⟪x, y⟫ • x, y⟫ * (t / ‖⟪x, y⟫‖)

Tactic:
·

Goal state after:
case e'_4.e'_6
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
hzero : ¬⟪x, y⟫ = 0
hzero' : ‖⟪x, y⟫‖ ≠ 0
⊢ 2 * ‖⟪x, y⟫‖ * t = 2 * re ⟪⟪x, y⟫ • x, y⟫ * (t / ‖⟪x, y⟫‖)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∫ (x : X) in support F, F x ∂ν = ∫ (x : X), F x ∂ν

Tactic:
nth_rw 2 [← setIntegral_univ]

Goal state after:
X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∫ (x : X) in support F, F x ∂ν = ∫ (x : X) in univ, F x ∂ν
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
M : Type u_5
inst✝² : NormedAddCommGroup M
inst✝¹ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
inst✝ : TopologicalSpace X
⊢ ∫ (x : X) in tsupport F, F x ∂ν = ∫ (x : X), F x ∂ν

Tactic:
nth_rw 2 [← setIntegral_univ]

Goal state after:
X : Type u_1
M : Type u_5
inst✝² : NormedAddCommGroup M
inst✝¹ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
inst✝ : TopologicalSpace X
⊢ ∫ (x : X) in tsupport F, F x ∂ν = ∫ (x : X) in univ, F x ∂ν
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
inst✝¹ : MeasurableSpace X
inst✝ : PseudoEMetricSpace X
μ : Measure X
E : Set X
E_mble : MeasurableSet E
δ : ℝ
⊢ μ E = lintegral μ (E.indicator fun x ↦ 1)

pp: X : Type u_1
inst✝¹ : MeasurableSpace X
inst✝ : PseudoEMetricSpace X
μ : Measure X
E : Set X
E_mble : MeasurableSet E
δ : ℝ
⊢ lintegral μ (E.indicator fun x ↦ 1) ≤ lintegral μ (thickenedIndicatorAux δ E)

Tactic:
·

Goal state after:
X : Type u_1
inst✝¹ : MeasurableSpace X
inst✝ : PseudoEMetricSpace X
μ : Measure X
E : Set X
E_mble : MeasurableSet E
δ : ℝ
⊢ μ E = lintegral μ (E.indicator fun x ↦ 1)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
inst✝¹ : MeasurableSpace X
inst✝ : PseudoEMetricSpace X
μ : Measure X
E : Set X
E_mble : MeasurableSet E
δ : ℝ
⊢ lintegral μ (E.indicator fun x ↦ 1) ≤ lintegral μ (thickenedIndicatorAux δ E)

Tactic:
·

Goal state after:
X : Type u_1
inst✝¹ : MeasurableSpace X
inst✝ : PseudoEMetricSpace X
μ : Measure X
E : Set X
E_mble : MeasurableSet E
δ : ℝ
⊢ lintegral μ (E.indicator fun x ↦ 1) ≤ lintegral μ (thickenedIndicatorAux δ E)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
hx : x ∈ s
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x

pp: case neg
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
hx : x ∉ s
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x

Tactic:
·

Goal state after:
case pos
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
hx : x ∈ s
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
hx : x ∉ s
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x

Tactic:
·

Goal state after:
case neg
X : Type u_6
f : X → ℝ
m0 : MeasurableSpace X
μ : Measure X
g : SimpleFunc X ℝ
hf : Integrable f μ
c : ℝ
s : Set X
hs : MeasurableSet s
x : X
hx : x ∉ s
⊢ (s.indicator (const X c) * f) x = s.indicator (c • f) x
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ √(‖r‖ ^ 2) * √(normSq x) ≤ ‖r‖ * ‖x‖

pp: case hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ 0 ≤ ‖r‖ ^ 2

Tactic:
·

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ √(‖r‖ ^ 2) * √(normSq x) ≤ ‖r‖ * ‖x‖
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: case hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ 0 ≤ ‖r‖ ^ 2

Tactic:
·

Goal state after:
case hx
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
r : 𝕜
x : F
⊢ 0 ≤ ‖r‖ ^ 2