mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs (x (n + k)) a‖ ≤ bound a

pp: case refine_3
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs (x (n + k)) a) atTop (𝓝 (f a))

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs (x (n + k)) a‖ ≤ bound a
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : 𝓝[>] c = ⊥
⊢ rightLim f c ∈ id s

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
⊢ rightLim f c ∈ id s

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : 𝓝[>] c = ⊥
⊢ rightLim f c ∈ id s
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
h''c : ¬∃ y, Tendsto f (𝓝[>] c) (𝓝 y)
⊢ rightLim f c ∈ id s

pp: case neg
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
h''c : ∃ y, Tendsto f (𝓝[>] c) (𝓝 y)
⊢ rightLim f c ∈ id s

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
a : α
h : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : (𝓝[>] c).NeBot
h''c : ¬∃ y, Tendsto f (𝓝[>] c) (𝓝 y)
⊢ rightLim f c ∈ id s
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_3
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs (x (n + k)) a) atTop (𝓝 (f a))

Tactic:
·

Goal state after:
case refine_3
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
l : Filter ι
inst✝ : l.IsCountablyGenerated
fs : ι → α → E
f : α → E
bound : α → ℝ
hfs_meas : ∀ᶠ (n : ι) in l, AEStronglyMeasurable (fs n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s
h✝ : {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x} ∈ l
k : ℕ
h : ∀ b ≥ k, x b ∈ {x | (fun n ↦ AEStronglyMeasurable (fs n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a) x}
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ fs (x (n + k)) a) atTop (𝓝 (f a))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : T3Space β
inst✝ : NoTopOrder α
f g : α → β
b : β
h : Tendsto f atTop (𝓝 b)
h' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f
hα : IsEmpty α
⊢ Tendsto g atTop (𝓝 b)

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : T3Space β
inst✝ : NoTopOrder α
f g : α → β
b : β
h : Tendsto f atTop (𝓝 b)
h' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f
hα : Nonempty α
⊢ Tendsto g atTop (𝓝 b)

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : T3Space β
inst✝ : NoTopOrder α
f g : α → β
b : β
h : Tendsto f atTop (𝓝 b)
h' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f
hα : IsEmpty α
⊢ Tendsto g atTop (𝓝 b)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hμ
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s ∩ t))

pp: case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))

Tactic:
·

Goal state after:
case hμ
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s ∩ t))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))

Tactic:
·

Goal state after:
case hν
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht : NullMeasurableSet t μ
hfs : IntegrableOn f s μ
⊢ Integrable f (μ.restrict (s \ t))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : OrderTop α
⊢ Tendsto (rightLim f) atTop (𝓝 b)

pp: case inr
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : NoTopOrder α
⊢ Tendsto (rightLim f) atTop (𝓝 b)

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : OrderTop α
⊢ Tendsto (rightLim f) atTop (𝓝 b)
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : NoTopOrder α
⊢ Tendsto (rightLim f) atTop (𝓝 b)

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : T3Space β
f : α → β
b : β
h : Tendsto f atTop (𝓝 b)
val✝ : NoTopOrder α
⊢ Tendsto (rightLim f) atTop (𝓝 b)
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] x = ⊥
⊢ leftLim f x ≤ f y

pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ leftLim f x ≤ f y

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] x = ⊥
⊢ leftLim f x ≤ f y
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr.refine_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ (f '' Iio x).Nonempty

pp: case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ ∀ b ∈ f '' Iio x, b ≤ f y

Tactic:
·

Goal state after:
case inr.refine_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ (f '' Iio x).Nonempty
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ ∀ b ∈ f '' Iio x, b ≤ f y

Tactic:
·

Goal state after:
case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ ∀ b ∈ f '' Iio x, b ≤ f y
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ f x ≤ leftLim f y

pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] y).NeBot
⊢ f x ≤ leftLim f y

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ f x ≤ leftLim f y
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x : α
h : x ≤ x
⊢ leftLim f x ≤ leftLim f x

pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
hxy : x < y
⊢ leftLim f x ≤ leftLim f y

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x : α
h : x ≤ x
⊢ leftLim f x ≤ leftLim f x
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
hxy : x < y
⊢ leftLim f x ≤ leftLim f y

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
hxy : x < y
⊢ leftLim f x ≤ leftLim f y
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ rightLim f x ≤ leftLim f y

pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] y).NeBot
⊢ rightLim f x ≤ leftLim f y

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ rightLim f x ≤ leftLim f y
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in (SummationFilter.unconditional ι).filter, AEStronglyMeasurable (fun a ↦ ∑ i ∈ n, F i a) μ

pp: case refine_2
α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in (SummationFilter.unconditional ι).filter, ∀ᵐ (a : α) ∂μ, ‖∑ i ∈ n, F i a‖ ≤ ∑' (n : ι), bound n a

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
E : Type u_2
F✝ : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F✝
inst✝¹ : NormedSpace ℝ F✝
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in (SummationFilter.unconditional ι).filter, AEStronglyMeasurable (fun a ↦ ∑ i ∈ n, F i a) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case insert
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ (x : X) in s a✝, f x ∂μ + ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ insert a✝ s✝, ∫ (x : X) in s i, f x ∂μ

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ Disjoint (s a✝) (⋃ i ∈ s✝, s i)

pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ MeasurableSet (⋃ i ∈ s✝, s i)

Tactic:
·

Goal state after:
case insert
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
s : ι → Set X
a✝ : ι
s✝ : Finset ι
hat : a✝ ∉ s✝
IH :
  (∀ i ∈ s✝, MeasurableSet (s i)) →
    (↑s✝).Pairwise (Disjoint on s) →
      (∀ i ∈ s✝, IntegrableOn f (s i) μ) → ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ s✝, ∫ (x : X) in s i, f x ∂μ
hs : MeasurableSet (s a✝) ∧ ∀ x ∈ s✝, MeasurableSet (s x)
hf : IntegrableOn f (s a✝) μ ∧ ∀ x ∈ s✝, IntegrableOn f (s x) μ
h's : (↑s✝).Pairwise (Disjoint on s) ∧ ∀ b ∈ ↑s✝, a✝ ≠ b → (Disjoint on s) a✝ b ∧ (Disjoint on s) b a✝
⊢ ∫ (x : X) in s a✝, f x ∂μ + ∫ (x : X) in ⋃ i ∈ s✝, s i, f x ∂μ = ∑ i ∈ insert a✝ s✝, ∫ (x : X) in s i, f x ∂μ