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

Goal state before:
pp: case pos
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : Countable ι
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSingletonClass X
x : ι → X
c : ι → ℝ≥0∞
inst✝ : FiniteDimensional ℝ E
hc : ∀ (i : ι), c i ≠ ∞
hf : Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))
⊢ (∫ (x : X), f x ∂Measure.sum fun i ↦ c i • Measure.dirac (x i)) = ∑' (i : ι), (c i).toReal • f (x i)

pp: case neg
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : Countable ι
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSingletonClass X
x : ι → X
c : ι → ℝ≥0∞
inst✝ : FiniteDimensional ℝ E
hc : ∀ (i : ι), c i ≠ ∞
hf : ¬Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))
⊢ (∫ (x : X), f x ∂Measure.sum fun i ↦ c i • Measure.dirac (x i)) = ∑' (i : ι), (c i).toReal • f (x i)

Tactic:
· rw [integral_sum_measure hf]
    congr with i
    rw [integral_smul_measure, integral_dirac]

Goal state after:
case neg
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁴ : Countable ι
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
f : X → E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSingletonClass X
x : ι → X
c : ι → ℝ≥0∞
inst✝ : FiniteDimensional ℝ E
hc : ∀ (i : ι), c i ≠ ∞
hf : ¬Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))
⊢ (∫ (x : X), f x ∂Measure.sum fun i ↦ c i • Measure.dirac (x i)) = ∑' (i : ι), (c i).toReal • f (x i)
Module: Mathlib.Topology.Algebra.Indicator

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
inst✝ : One β
hs : ∀ a ∈ frontier s, f a = 1
hf : ContinuousOn f (closure[inst✝²] s)
⊢ Continuous[inst✝², inst✝¹] (s.mulIndicator f)

Tactic:
classical exact continuous_piecewise hs hf continuousOn_const

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
inst✝¹ : PseudoMetricSpace R
inst✝ : Sub R
K : ℝ≥0
lip : LipschitzWith K fun p ↦ p.1 - p.2
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
⊢ Bornology.IsBounded (s - t)

Tactic:
have bdd : Bornology.IsBounded (s ×ˢ t) := Bornology.IsBounded.prod s_bdd t_bdd

Goal state after:
R : Type u_1
inst✝¹ : PseudoMetricSpace R
inst✝ : Sub R
K : ℝ≥0
lip : LipschitzWith K fun p ↦ p.1 - p.2
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
bdd : Bornology.IsBounded (s ×ˢ t)
⊢ Bornology.IsBounded (s - t)
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
inst✝² : PseudoMetricSpace R
inst✝¹ : Monoid R
inst✝ : LipschitzMul R
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
⊢ Bornology.IsBounded (s * t)

Tactic:
have bdd : Bornology.IsBounded (s ×ˢ t) := Bornology.IsBounded.prod s_bdd t_bdd

Goal state after:
R : Type u_1
inst✝² : PseudoMetricSpace R
inst✝¹ : Monoid R
inst✝ : LipschitzMul R
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
bdd : Bornology.IsBounded (s ×ˢ t)
⊢ Bornology.IsBounded (s * t)
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: E' : Type u_1
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
⊢ 0 < lipschitzExtensionConstant E'

Tactic:
rw [lipschitzExtensionConstant]

Goal state after:
E' : Type u_1
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
⊢ 0 <
    have A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv;
    max (‖↑A.symm‖₊ * ‖↑A‖₊) 1
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s

Tactic:
refine ⟨A.symm ∘ g, ?_, ?_⟩

Goal state after:
case refine_1
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ LipschitzWith (lipschitzExtensionConstant E' * K) (⇑A.symm ∘ g)

case refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
⊢ EqOn f (⇑A.symm ∘ g) s
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
hF : CompleteSpace F
hfi : Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T hT g

Tactic:
have hgi : Integrable g μ := hfi.congr h
    rw [setToFun_eq hT hfi, setToFun_eq hT hgi, (Integrable.toL1_eq_toL1_iff f g hfi hgi).2 h]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
x : α
hx : x ∈ s
⊢ f x = (⇑A.symm ∘ g) x

Tactic:
have : A (f x) = g x := gs hx

Goal state after:
case refine_2
α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
A : E' ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ E') → ℝ := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
LA : LipschitzWith ‖↑A‖₊ ⇑A
L : LipschitzOnWith (‖↑A‖₊ * K) (⇑A ∘ f) s
g : α → ι → ℝ
hg : LipschitzWith (‖↑A‖₊ * K) g
gs : EqOn (⇑A ∘ f) g s
x : α
hx : x ∈ s
this : A (f x) = g x
⊢ f x = (⇑A.symm ∘ g) x
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace F
inst✝ : IsFiniteMeasure μ
hT : DominatedFinMeasAdditive μ T C
x : E
⊢ (setToFun μ T hT fun x_1 ↦ x) = (T univ) x

Tactic:
have : (fun _ : α => x) = Set.indicator univ fun _ => x := (indicator_univ _).symm

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace F
inst✝ : IsFiniteMeasure μ
hT : DominatedFinMeasAdditive μ T C
x : E
this : (fun x_1 ↦ x) = univ.indicator fun x_1 ↦ x
⊢ (setToFun μ T hT fun x_1 ↦ x) = (T univ) x
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
⊢ Continuous[Pi.topologicalSpace, topologicalSpace]
    (LinearMap.toContinuousLinearMap.symm ≪≫ₗ LinearEquiv.piRing 𝕜 E ι 𝕜).invFun

Tactic:
simp_rw [LinearEquiv.invFun_eq_symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]

Goal state after:
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
⊢ Continuous[Pi.topologicalSpace, topologicalSpace]
    ⇑((LinearEquiv.piRing 𝕜 E ι 𝕜).symm ≪≫ₗ LinearMap.toContinuousLinearMap)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case inl
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_nonneg : 0 ≤ᵐ[μ] f
hf_int : Integrable f μ
ε : ℝ
hμ : μ {x | ε ≤ f x} = ∞
⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ

Tactic:
simpa [measureReal_def, hμ] using integral_nonneg_of_ae hf_nonneg

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1

Tactic:
by_cases hx : x ∈ s <;> by_cases hy : y ∈ s

Goal state after:
case pos
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
hx : x ∈ s
hy : y ∈ s
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1

case neg
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
hx : x ∈ s
hy : y ∉ s
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1

case pos
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
hx : x ∉ s
hy : y ∈ s
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1

case neg
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
hx : x ∉ s
hy : y ∉ s
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f : α →ᵇ β
n : ℕ
⊢ ⇑(nsmulRec n f + f) = n • ⇑f + ⇑f

Tactic:
coe_add,

Goal state after:
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f : α →ᵇ β
n : ℕ
⊢ ⇑(nsmulRec n f) + ⇑f = n • ⇑f + ⇑f
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
V : Type u_1
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : T2Space V
inst✝³ : Nontrivial V
inst✝² : LocallyCompactSpace V
inst✝¹ : Module 𝕜 V
inst✝ : ContinuousSMul 𝕜 V
⊢ LocallyCompactSpace 𝕜

Tactic:
by
    obtain ⟨v, hv⟩ : ∃ v : V, v ≠ 0 := exists_ne 0
    let L : 𝕜 → V := fun t ↦ t • v
    have : IsClosedEmbedding L := isClosedEmbedding_smul_left hv
    apply IsClosedEmbedding.locallyCompactSpace this

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u_1
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ
x : α
⊢ (fun f ↦ f.toFun) 0 x ≤ (fun f ↦ f.toFun) (const α ‖f‖ - f) x

Tactic:
simp only [ContinuousMap.toFun_eq_coe, coe_toContinuousMap, coe_zero, Pi.zero_apply, coe_sub,
    const_apply, Pi.sub_apply]

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
f : α →ᵇ ℝ
x : α
⊢ 0 ≤ ‖f‖ - f x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ ∫ (x : β), f x ∂μ = ∑ x ∈ f.range, μ.real (⇑f ⁻¹' {x}) • x

Tactic:
← f.coe_toLargerSpace_eq hm

Goal state after:
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β →ₛ F
hf_int : Integrable (⇑f) μ
⊢ ∫ (x : β), (SimpleFunc.toLargerSpace hm f) x ∂μ = ∑ x ∈ f.range, μ.real (⇑(SimpleFunc.toLargerSpace hm f) ⁻¹' {x}) • x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : ¬Integrable f μ
hf_int_m : ¬Integrable f (μ.trim hm)
⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm

Tactic:
integral_undef hf_int,

Goal state after:
case neg
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
β : Type u_6
m m0 : MeasurableSpace β
μ : Measure β
hm : m ≤ m0
f : β → G
hf : StronglyMeasurable f
hG : CompleteSpace G
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hf_int : ¬Integrable f μ
hf_int_m : ¬Integrable f (μ.trim hm)
⊢ 0 = ∫ (x : β), f x ∂μ.trim hm
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
⊢ μ s < ∞ → setToFun μ T hT (s.indicator fun x ↦ c) = setToFun μ' T hT' (s.indicator fun x ↦ c)

Tactic:
hμs

Goal state after:
case pos.h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ setToFun μ T hT (s.indicator fun x ↦ c) = setToFun μ' T hT' (s.indicator fun x ↦ c)
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ 2 * μ univ * ↑r

Tactic:
this,

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ : ¬IsFiniteMeasure μ
this : μ univ = ∞
⊢ eLpNorm f 1 μ ≤ 2 * ∞ * ↑r
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.h_ae
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
⊢ ∀ ⦃f g : α → E⦄,
    f =ᵐ[μ] g → Integrable f μ → setToFun μ T hT f = setToFun μ' T hT' f → setToFun μ T hT g = setToFun μ' T hT' g

Tactic:
intro f₂ g₂ hfg _ hf_eq
    have hfg' : f₂ =ᵐ[μ'] g₂ := (Measure.absolutelyContinuous_of_le_smul hμ'_le).ae_eq hfg
    rw [← setToFun_congr_ae hT hfg, hf_eq, setToFun_congr_ae hT' hfg']

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T T' : Set α → E →L[ℝ] F
C C' : ℝ
f : α → E
ν : Measure α
hTμ : DominatedFinMeasAdditive μ T C
hTν : DominatedFinMeasAdditive ν T' C'
hμ : Integrable f μ
hν : Integrable f ν
hTμ_add : DominatedFinMeasAdditive (μ + ν) T (max C 0)
hTν_add : DominatedFinMeasAdditive (μ + ν) T' (max C' 0)
⊢ setToFun (μ + ν) T hTμ_add f + setToFun (μ + ν) T' hTν_add f = setToFun μ T hTμ f + setToFun ν T' hTν f

Tactic:
by
      rw [setToFun_congr_measure_of_add_right hTμ_add hTμ f (hμ.add_measure hν),
        setToFun_congr_measure_of_add_left hTν_add hTν f (hμ.add_measure hν)]

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: 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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
⊢ Tendsto (leftLim f) (𝓝[≤] a) (𝓝 (leftLim f a))

Tactic:
this,

Goal state after:
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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
⊢ Tendsto (leftLim f) (𝓝[<] a ⊔ pure a) (𝓝 (leftLim f a))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: 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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
⊢ leftLim f c ∈ id s

Tactic:
rcases eq_or_neBot (𝓝[<] c) with h'c | h'c

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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
u : α
au : u < a
hu : Ioo u a ⊆ {x | f x ∈ s}
c : α
hc : c ∈ Ioo u a
h'c : 𝓝[<] c = ⊥
⊢ leftLim f c ∈ id s

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))
this : 𝓝[≤] a = 𝓝[<] a ⊔ pure a
s : Set β
s_mem : s ∈ 𝓝 (leftLim f a)
s_closed : IsClosed[inst✝³] s
h' : (𝓝[<] a).NeBot
b : α
hb : b ∈ Iio a
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
⊢ leftLim f c ∈ id s
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
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✝
⊢ ∀ i ∈ s✝, Disjoint (s a✝) (s i)

Tactic:
exact fun i hi => (h's.2 i hi (ne_of_mem_of_not_mem hi hat).symm).1

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set X
μ : Measure X
hs : ∀ᵐ (x : X) ∂μ, x ∈ s
f : X → E
⊢ s =ᵐ[μ] univ

Tactic:
rwa [ae_eq_univ]

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

Goal state before:
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
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
⊢ ∫ (x : X), (⋃ i ∈ t, s i).indicator f x ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ

Tactic:
this,

Goal state after:
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
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
⊢ ∫ (x : X), (⋃ i ∈ t, s i).indicator f x ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos
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
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
⊢ StronglyMeasurable fun x ↦ setToFun μ T hT (f x)

Tactic:
have : SeparableSpace (range (Function.uncurry f) ∪ {0} : Set E) :=
    hf.separableSpace_range_union_singleton

Goal state after:
case pos
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
β : Type u_7
mβ : MeasurableSpace β
inst✝ : SFinite μ
hT : DominatedFinMeasAdditive μ T C
h'T : ∀ (s : Set (β × α)), MeasurableSet s → StronglyMeasurable fun x ↦ T (Prod.mk x ⁻¹' s)
f : β → α → E
hf : StronglyMeasurable (Function.uncurry f)
hF : CompleteSpace F
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (Function.uncurry f) ∪ {0})
⊢ StronglyMeasurable fun x ↦ setToFun μ T hT (f x)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
⊢ MeasurableSet (Ioi c)

Tactic:
refine OuterMeasure.ofFunction_caratheodory fun t => ?_

Goal state after:
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
⊢ f.length (t ∩ Ioi c) + f.length (t \ Ioi c) ≤ f.length t