mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case h₂
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ ≤ ∏ i, ‖m (e.symm i)‖

pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ Fintype (Subtype (Membership.mem s))

Tactic:
]

Goal state after:
case h₂
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ ≤ ∏ i, ‖m (e.symm i)‖

𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ Fintype (Subtype (Membership.mem s))
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ Integrable (fun a ↦ K.indicator f a • g a) μ

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ Integrable (fun a ↦ K.indicator f a • g a) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : ι) → E₁ i
⊢ ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s } = ‖f‖ * ‖x‖ ^ Fintype.card { a // a ∉ s }

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: case hφ
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ IntegrableOn f K μ

Tactic:
]

Goal state after:
case hφ
X : Type u_1
E : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : TopologicalSpace X
inst✝⁵ : NormedAddCommGroup E
μ : Measure X
𝕜 : Type u_9
inst✝⁴ : NormedRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → 𝕜
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous[inst✝⁶, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
this : (K.indicator fun x ↦ f x • g x) = fun x ↦ f x • g x
⊢ IntegrableOn f K μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : Fintype ι
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : MultilinearMap 𝕜 E G
ε : ι → ℝ
C : ℝ
c : ι → 𝕜
hε : ∀ (i : ι), 0 < ε i
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m m✝ : (i : ι) → E i
i✝ : ι
h : ‖m✝ i✝‖ = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
⊢ MemLp f ∞ (μ.restrict s)

Tactic:
borelize E

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁵ : BorelSpace X
inst✝⁴ : ConditionallyCompleteLinearOrder X
inst✝³ : ConditionallyCompleteLinearOrder E
inst✝² : OrderTopology X
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
f : X → E
hmono : MonotoneOn f s
a b : X
ha : IsLeast s a
hb : IsGreatest s b
h's : MeasurableSet s
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ MemLp f ∞ (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ g x * f x) k μ

Tactic:
]

Goal state after:
X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ g x * f x) k μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ f x * g x) k μ

Tactic:
]

Goal state after:
X : Type u_1
R : Type u_8
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
μ : Measure X
inst✝⁴ : OpensMeasurableSpace X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsLocallyClosed s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ f x * g x) k μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → E
g : X → 𝕜
s : Set X
hs : IsLocallyClosed s
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ g x • f x) k μ

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → E
g : X → 𝕜
s : Set X
hs : IsLocallyClosed s
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ g x • f x) k μ
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
s : Set α
x✝ : MeasurableSet s
hμs : c ≠ 0 ∧ μ s = ∞ ∨ c = ∞ ∧ μ s ≠ 0
⊢ μ s = ∞

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
s : Set α
x✝ : MeasurableSet s
hμs : c ≠ 0 ∧ μ s = ∞ ∨ c = ∞ ∧ μ s ≠ 0
⊢ μ s = ∞
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_zero : c ≠ 0
hT : FinMeasAdditive μ T
s : Set α
x✝ : MeasurableSet s
hμs : μ s = ∞
⊢ c ≠ 0 ∧ μ s = ∞ ∨ c = ∞ ∧ μ s ≠ 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_zero : c ≠ 0
hT : FinMeasAdditive μ T
s : Set α
x✝ : MeasurableSet s
hμs : μ s = ∞
⊢ c ≠ 0 ∧ μ s = ∞ ∨ c = ∞ ∧ μ s ≠ 0
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → 𝕜
g : X → E
s : Set X
hs : IsLocallyClosed s
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ f x • g x) k μ

Tactic:
]

Goal state after:
X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
inst✝⁶ : OpensMeasurableSpace X
inst✝⁵ : LocallyCompactSpace X
inst✝⁴ : T2Space X
𝕜 : Type u_9
inst✝³ : NormedRing 𝕜
inst✝² : SecondCountableTopologyEither X E
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : X → 𝕜
g : X → E
s : Set X
hs : IsLocallyClosed s
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x ↦ f x • g x) k μ
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : AddCancelMonoid β
T : Set α → β
hT : T ∅ = T ∅ + T ∅
h_empty : μ ∅ ≠ ∞
⊢ T ∅ = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : AddCancelMonoid β
T : Set α → β
hT : T ∅ = T ∅ + T ∅
h_empty : μ ∅ ≠ ∞
⊢ T ∅ = 0
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : AddCancelMonoid β
T : Set α → β
hT : T ∅ + 0 = T ∅ + T ∅
h_empty : μ ∅ ≠ ∞
⊢ T ∅ = 0

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : AddCancelMonoid β
T : Set α → β
hT : T ∅ + 0 = T ∅ + T ∅
h_empty : μ ∅ ≠ ∞
⊢ T ∅ = 0
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
h_meas : Measurable fun x ↦ ‖f x - y₀‖
⊢ eLpNorm (fun x ↦ f x - y₀) p μ < ∞

Tactic:
]

Goal state after:
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
h_meas : Measurable fun x ↦ ‖f x - y₀‖
⊢ eLpNorm (fun x ↦ f x - y₀) p μ < ∞
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
hf' : MemLp (fun x ↦ ‖f x - y₀‖) p μ
x : β
⊢ ‖f x - y₀‖ + ‖f x - y₀‖ = ‖f x - y₀‖ + ‖f x - y₀‖

pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
hf' : MemLp (fun x ↦ ‖f x - y₀‖) p μ
x : β
⊢ 0 ≤ ‖f x - y₀‖ + ‖f x - y₀‖

Tactic:
]

Goal state after:
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
p : ℝ≥0∞
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f p μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) p μ
n : ℕ
hf' : MemLp (fun x ↦ ‖f x - y₀‖) p μ
x : β
⊢ 0 ≤ ‖f x - y₀‖ + ‖f x - y₀‖
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ g, eLpNorm (f - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ

Tactic:
borelize E

Goal state after:
β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ ∃ g, eLpNorm (f - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
this : eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ
⊢ eLpNorm (f' - ⇑g) p μ < ε

Tactic:
]

Goal state after:
β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
g : β →ₛ E
hg : eLpNorm (f' - ⇑g) p μ < ε
g_mem : MemLp (⇑g) p μ
this : eLpNorm (f - ⇑g) p μ = eLpNorm (f' - ⇑g) p μ
⊢ eLpNorm (f' - ⇑g) p μ < ε
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
hf' : MemLp f' p μ
f'meas : Measurable f'
⊢ ∃ g, eLpNorm (f' - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ

Tactic:
have : SeparableSpace (range f' ∪ {0} : Set E) :=
    StronglyMeasurable.separableSpace_range_union_singleton hf.1.stronglyMeasurable_mk

Goal state after:
β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
hf' : MemLp f' p μ
f'meas : Measurable f'
this : SeparableSpace ↑(Set.range f' ∪ {0})
⊢ ∃ g, eLpNorm (f' - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
hf' : MemLp f' p μ
f'meas : Measurable f'
this : SeparableSpace ↑(Set.range f' ∪ {0})
n : ℕ
hn : eLpNorm (f' - ⇑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 ⋯ n)) p μ < ε
⊢ ∃ g, eLpNorm (f' - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ

Tactic:
]

Goal state after:
β : Type u_2
inst✝¹ : MeasurableSpace β
p : ℝ≥0∞
E : Type u_7
inst✝ : NormedAddCommGroup E
f : β → E
μ : Measure β
hf : MemLp f p μ
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
f' : β → E := AEStronglyMeasurable.mk f ⋯
hf' : MemLp f' p μ
f'meas : Measurable f'
this : SeparableSpace ↑(Set.range f' ∪ {0})
n : ℕ
hn : eLpNorm (f' - ⇑(approxOn f' f'meas (Set.range f' ∪ {0}) 0 ⋯ n)) p μ < ε
⊢ ∃ g, eLpNorm (f' - ⇑g) p μ < ε ∧ MemLp (⇑g) p μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f 1 μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) 1 μ
n : ℕ
⊢ MemLp (⇑(approxOn f fmeas s y₀ h₀ n)) 1 μ

Tactic:
]

Goal state after:
β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
f : β → E
μ : Measure β
fmeas : Measurable f
hf : MemLp f 1 μ
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
hi₀ : MemLp (fun x ↦ y₀) 1 μ
n : ℕ
⊢ MemLp (⇑(approxOn f fmeas s y₀ h₀ n)) 1 μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
F : Type u_5
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup F
p : ℝ
f : α →ₛ F
μ : Measure α
h_map : (fun x ↦ ‖f x‖ₑ ^ p) = ⇑(map (fun x ↦ ‖x‖ₑ ^ p) f)
⊢ (∑ x ∈ f.range, ‖x‖ₑ ^ p * μ (⇑f ⁻¹' {x})) ^ (1 / p) = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (⇑f ⁻¹' {y})) ^ (1 / p)

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
⊢ ∫⁻ (a : α), (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f) a ∂μ ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
⊢ ∫⁻ (a : α), (map (fun x ↦ ‖x‖ₑ ^ p.toReal) f) a ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
h_fin : f.FinMeasSupp μ
this : ∀ {b : E}, (fun x ↦ ‖x‖ₑ ^ p.toReal) b = 0 ↔ b = 0
⊢ f.FinMeasSupp μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
hp_pos : p ≠ 0
hp_ne_top : p ≠ ∞
f : α →ₛ E
hf : MemLp (⇑f) p μ
y : E
hy_ne : y ≠ 0
h_fin : f.FinMeasSupp μ
this : ∀ {b : E}, (fun x ↦ ‖x‖ₑ ^ p.toReal) b = 0 ↔ b = 0
⊢ f.FinMeasSupp μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : p = 0
⊢ AEStronglyMeasurable (⇑f) μ

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : p = 0
⊢ AEStronglyMeasurable (⇑f) μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : p = ∞
⊢ MemLp ⇑f ∞ μ

Tactic:
]

Goal state after:
case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : p = ∞
⊢ MemLp ⇑f ∞ μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ (∑ y ∈ f.range, ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y})) ^ (1 / p.toReal) < ∞

Tactic:
]

Goal state after:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
f : α →ₛ E
hf : ∀ (y : E), y ≠ 0 → μ (⇑f ⁻¹' {y}) < ∞
hp0 : ¬p = 0
hp_top : ¬p = ∞
⊢ (∑ y ∈ f.range, ‖y‖ₑ ^ p.toReal * μ (⇑f ⁻¹' {y})) ^ (1 / p.toReal) < ∞