mathlib-initiative/mathlib-tactics huggingface dataset:
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
l u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
h✝ : Nonempty R
⊢ f.measure univ < ∞

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
l u : ℝ
hfl : Tendsto (↑f) atBot (𝓝 l)
hfu : Tendsto (↑f) atTop (𝓝 u)
h✝ : Nonempty R
⊢ f.measure univ < ∞
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : IsEmpty R
⊢ IsFiniteMeasure f.measure

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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
⊢ IsFiniteMeasure f.measure

Tactic:
·

Goal state after:
case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : IsEmpty R
⊢ IsFiniteMeasure f.measure
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : Tendsto (↑f) atTop atTop
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : ∃ l, Tendsto (↑f) atTop (𝓝 l)
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)

Tactic:
·

Goal state after:
case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
H : Tendsto (↑f) atTop atTop
⊢ ∃ u, Tendsto (↑f) atTop (𝓝 u)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀

pp: case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ ⋂ i ∈ t, clPrehaar (↑K₀) i

Tactic:
·

Goal state after:
case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : Tendsto (↑f) atBot atBot
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)

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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : ∃ l, Tendsto (↑f) atBot (𝓝 l)
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)

Tactic:
·

Goal state after:
case inl
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
C : ℝ
h : ∀ (x : R), |↑f x| ≤ C
h✝ : Nonempty R
u : ℝ
hu : Tendsto (↑f) atTop (𝓝 u)
H : Tendsto (↑f) atBot atBot
⊢ ∃ l, Tendsto (↑f) atBot (𝓝 l)
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ ⋂ i ∈ t, clPrehaar (↑K₀) i

Tactic:
·

Goal state after:
case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋯
h1V₀ : IsOpen[inst✝¹] V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ ⋂ i ∈ t, clPrehaar (↑K₀) i
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ (a : X), ENNReal.ofReal (g a) ≤ 1

pp: case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0

Tactic:
·

Goal state after:
case pos.hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ (a : X), ENNReal.ofReal (g a) ≤ 1
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∈ s
⊢ g x ≤ ENNReal.toReal 1

pp: case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∉ s
⊢ g x ≤ ENNReal.toReal 1

Tactic:
·

Goal state after:
case pos
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∈ s
⊢ g x ≤ ENNReal.toReal 1
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x

pp: case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

pp: case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

Tactic:
·

Goal state after:
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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ↑f x - l = ↑g x - l
⊢ ↑f x = ↑g x
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
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∉ s
⊢ g x ≤ ENNReal.toReal 1

Tactic:
·

Goal state after:
case neg
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H✝ : Integrable f μ
g : X → ℝ := fun x ↦ max (f x) 0
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
x : X
H : x ∉ s
⊢ g x ≤ ENNReal.toReal 1
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l

pp: case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

Tactic:
·

Goal state after:
case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑g x - l
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l

Tactic:
·

Goal state after:
case 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
l : ℝ
hfg : f.measure = g.measure
hfl : Tendsto (↑f) atBot (𝓝 l)
hgl : Tendsto (↑g) atBot (𝓝 l)
x : R
hf : ofReal (↑g x - l) = ofReal (↑f x - l)
⊢ 0 ≤ ↑f x - l
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0

Tactic:
·

Goal state after:
case pos.h'f
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
f : X → ℝ
s : Set X
hs : ∀ x ∈ s, f x ≤ 1
h's : ∀ x ∈ sᶜ, f x ≤ 0
H : Integrable f μ
g : X → ℝ := ⋯
g_int : Integrable g μ
this : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)
⊢ ∀ a ∈ sᶜ, ENNReal.ofReal (g a) = 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
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 : ↑g y - ↑f x = ↑g y - ↑g x
⊢ ↑f x = ↑g x

pp: case inl.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 : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ 0 ≤ ↑g y - ↑g x

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)
⊢ 0 ≤ ↑f y - ↑f x

Tactic:
·

Goal state after:
case inl
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 : ↑g y - ↑f x = ↑g y - ↑g x
⊢ ↑f x = ↑g x
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl.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 : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ 0 ≤ ↑g y - ↑g x

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)
⊢ 0 ≤ ↑f y - ↑f x

Tactic:
·

Goal state after:
case inl.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 : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ 0 ≤ ↑g y - ↑g x
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)
⊢ 0 ≤ ↑f y - ↑f x

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)
⊢ 0 ≤ ↑f y - ↑f x
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)
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

pp: case pos
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)
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (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)
⊢ ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ
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)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → 0 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 | 0 x ≤ g x} (μ.restrict s)

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)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ s → 0 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 | 0 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 | 0 x ≤ g x} (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.refine_1
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 ≤ᵐ[μ.restrict s] f

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)
⊢ f ≤ᵐ[μ.restrict s] g

Tactic:
·

Goal state after:
case pos.refine_1
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 ≤ᵐ[μ.restrict s] f
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case pos.refine_1
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 → 0 x ≤ f 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 | 0 x ≤ f x} (μ.restrict s)

Tactic:
·

Goal state after:
case pos.refine_1
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 → 0 x ≤ f 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 | 0 x ≤ f 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 | 0 x ≤ f x} (μ.restrict s)
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}
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)
⊢ f ≤ᵐ[μ.restrict s] g

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)
⊢ f ≤ᵐ[μ.restrict s] g