mathlib-initiative/mathlib-tactics huggingface dataset:
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 T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : α → E
hF : CompleteSpace F
hf : Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T' hT' f

pp: case neg
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
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T' hT' f

Tactic:
· simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h]

Goal state after:
case neg
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
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
h : T = T'
f : α → E
hF : CompleteSpace F
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T' hT' f
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 : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i) μ
⊢ (setToFun μ T hT fun a ↦ ∑ i ∈ s, f i a) = ∑ i ∈ s, setToFun μ T hT (f i)

Tactic:
convert! setToFun_finsetSum' hT s hf with a

Goal state after:
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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i) μ
a : α
⊢ ∑ i ∈ s, f i a = (∑ i ∈ s, f i) a
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: case h
R : Type u_1
inst✝ : NonUnitalSeminormedRing R
s t : Set R
hs : ∃ C, ∀ ⦃x : R⦄, x ∈ s → ∀ ⦃y : R⦄, y ∈ s → dist x y ≤ C
ht : ∃ C, ∀ ⦃x : R⦄, x ∈ t → ∀ ⦃y : R⦄, y ∈ t → dist x y ≤ C
Af : ℝ
hAf : s ⊆ Metric.closedBall 0 Af
Ag : ℝ
hAg : t ⊆ Metric.closedBall 0 Ag
z : R
hz : z ∈ s * t
w : R
hw : w ∈ s * t
x₁ : R
hx₁ : x₁ ∈ s
y₁ : R
hy₁ : y₁ ∈ t
z_eq : x₁ * y₁ = z
⊢ dist z w ≤ 2 * Af * Ag

Tactic:
obtain ⟨x₂, hx₂, y₂, hy₂, w_eq⟩ := Set.mem_mul.mp hw

Goal state after:
case h
R : Type u_1
inst✝ : NonUnitalSeminormedRing R
s t : Set R
hs : ∃ C, ∀ ⦃x : R⦄, x ∈ s → ∀ ⦃y : R⦄, y ∈ s → dist x y ≤ C
ht : ∃ C, ∀ ⦃x : R⦄, x ∈ t → ∀ ⦃y : R⦄, y ∈ t → dist x y ≤ C
Af : ℝ
hAf : s ⊆ Metric.closedBall 0 Af
Ag : ℝ
hAg : t ⊆ Metric.closedBall 0 Ag
z : R
hz : z ∈ s * t
w : R
hw : w ∈ s * t
x₁ : R
hx₁ : x₁ ∈ s
y₁ : R
hy₁ : y₁ ∈ t
z_eq : x₁ * y₁ = z
x₂ : R
hx₂ : x₂ ∈ s
y₂ : R
hy₂ : y₂ ∈ t
w_eq : x₂ * y₂ = w
⊢ dist z w ≤ 2 * Af * Ag
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
hgi : Integrable g μ
⊢ setToFun μ T hT f = setToFun μ T hT g

Tactic:
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_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)

Tactic:
have LAsymm : LipschitzWith ‖A.symm.toContinuousLinearMap‖₊ A.symm := by
      apply A.symm.lipschitz
    apply (LAsymm.comp hg).weaken
    rw [lipschitzExtensionConstant, ← mul_assoc]
    exact mul_le_mul' (le_max_left _ _) le_rfl

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

Goal state before:
pp: case inr
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} < ∞
this : Fact (μ {x | ε ≤ f x} < ∞)
⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ

Tactic:
calc
      ε * μ.real { x | ε ≤ f x } = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
      _ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
        integral_mono_ae (integrable_const _) (hf_int.mono_measure μ.restrict_le_self) <|
          ae_restrict_mem₀ <| hf_int.aemeasurable.nullMeasurable measurableSet_Ici
      _ ≤ _ := integral_mono_measure μ.restrict_le_self hf_nonneg hf_int

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

Goal state before:

Tactic:
dist (F x) (F y) ≤ dist (f 0 x) (f 0 y) + (dist (f 0 x) (F x) + dist (f 0 y) (F y)) :=
          dist_triangle4_left _ _ _ _
        _ ≤ C + (b 0 + b 0) := add_le_add (hC x y) (add_le_add (fF_bdd x 0) (fF_bdd y 0))

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 : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
hl : l.NeBot
hfi : Integrable f μ
f_lp : ↥(Lp E 1 μ) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1 μ) := fun i ↦ if hFi : Integrable (fs i) μ then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i ↦ setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i) μ
⊢ ↑↑(F_lp i) =ᵐ[μ] fs i

Tactic:
simp_rw [F_lp, dif_pos hi]

Goal state after:
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
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
f : α → E
hf : AEStronglyMeasurable f μ
fs : ι → α → E
l : Filter ι
hfsi : ∀ᶠ (i : ι) in l, Integrable (fs i) μ
hfs : Tendsto (fun i ↦ ∫⁻ (x : α), ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)
hl : l.NeBot
hfi : Integrable f μ
f_lp : ↥(Lp E 1 μ) := Integrable.toL1 f hfi
F_lp : ι → ↥(Lp E 1 μ) := fun i ↦ if hFi : Integrable (fs i) μ then Integrable.toL1 (fs i) hFi else 0
tendsto_L1 : Tendsto F_lp l (𝓝 f_lp)
this : Tendsto (fun i ↦ setToFun μ T hT ↑↑(F_lp i)) l (𝓝 (setToFun μ T hT f))
i : ι
hi : Integrable (fs i) μ
⊢ ↑↑(Integrable.toL1 (fs i) hi) =ᵐ[μ] fs i
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖star f‖ ≤ ‖f‖

Tactic:
← BoundedContinuousFunction.norm_mkOfCompact,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : SeminormedAddCommGroup β
inst✝² : StarAddMonoid β
inst✝¹ : NormedStarGroup β
inst✝ : CompactSpace α
f : C(α, β)
⊢ ‖mkOfCompact (star f)‖ ≤ ‖f‖
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: 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
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
this : -f =ᵐ[μ] 0
ω : α
⊢ (-f) ω = 0 ω → f ω = 0 ω

Tactic:
hω

Goal state after:
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
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
this : -f =ᵐ[μ] 0
ω : α
hω : (-f) ω = 0 ω
⊢ f ω = 0 ω
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: 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
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
ω : α
⊢ f ω ≤ 0 → 0 ω ≤ (-f) ω

Tactic:
hω

Goal state after:
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
hr : r = 0
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ 0
hnegf : ∫ (x : α), -f x ∂μ = 0
ω : α
hω : f ω ≤ 0
⊢ 0 ω ≤ (-f) ω
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.h_add
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'
f₂ g₂ : α → E
⊢ Disjoint (Function.support f₂) (Function.support g₂) →
    Integrable f₂ μ →
      Integrable g₂ μ →
        setToFun μ T hT f₂ = setToFun μ' T hT' f₂ →
          setToFun μ T hT g₂ = setToFun μ' T hT' g₂ → setToFun μ T hT (f₂ + g₂) = setToFun μ' T hT' (f₂ + g₂)

Tactic:
_

Goal state after:
case pos.h_add
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'
f₂ g₂ : α → E
a✝ : Disjoint (Function.support f₂) (Function.support g₂)
⊢ Integrable f₂ μ →
    Integrable g₂ μ →
      setToFun μ T hT f₂ = setToFun μ' T hT' f₂ →
        setToFun μ T hT g₂ = setToFun μ' T hT' g₂ → setToFun μ T hT (f₂ + g₂) = setToFun μ' T hT' (f₂ + g₂)
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'
f₂ g₂ : α → E
hfg : f₂ =ᵐ[μ] g₂
⊢ Integrable f₂ μ → setToFun μ T hT f₂ = setToFun μ' T hT' f₂ → setToFun μ T hT g₂ = setToFun μ' T hT' g₂

Tactic:
_

Goal state after:
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'
f₂ g₂ : α → E
hfg : f₂ =ᵐ[μ] g₂
a✝ : Integrable f₂ μ
⊢ setToFun μ T hT f₂ = setToFun μ' T hT' f₂ → setToFun μ T hT g₂ = setToFun μ' T hT' g₂
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
ω : α
⊢ f ω ≤ ↑r → max (f ω) 0 ≤ ↑r

Tactic:
hω

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ≥0
hfint : Integrable f μ
hfint' : ∫ (a : α), ↑(-f a).toNNReal ∂μ ≤ ∫ (a : α), ↑(f a).toNNReal ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r
hr : ¬r = 0
hμ this : IsFiniteMeasure μ
ω : α
hω : f ω ≤ ↑r
⊢ max (f ω) 0 ≤ ↑r
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 μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
⊢ setToFun μ'' T'' hT'' f = setToFun μ T hT f + setToFun μ' T' hT' f

Tactic:
have I' : DominatedFinMeasAdditive (μ + μ') T' C' := .add_measure_left _ _ hT' hC'

Goal state after:
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 μ' μ'' : Measure α
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T' C'
hT'' : DominatedFinMeasAdditive μ'' T'' C''
h : ∀ (s : Set α), MeasurableSet s → (μ + μ') s < ∞ → T'' s = T s + T' s
hf : Integrable f μ
hf' : Integrable f μ'
hμ : μ'' ≤ μ + μ'
hC : 0 ≤ C
hC' : 0 ≤ C'
hC'' : 0 ≤ C''
I : DominatedFinMeasAdditive (μ + μ') T C
A : setToFun (μ + μ') T I f = setToFun μ T hT f
I' : DominatedFinMeasAdditive (μ + μ') T' C'
⊢ setToFun μ'' T'' hT'' f = setToFun μ T hT f + setToFun μ' T' hT' f
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
obs : ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ thickening δ E
δ : ℝ
x_notin_thE : x ∉ thickening δ E
⊢ δ ∈ Ioi 0 →
    (thickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
_

Goal state after:
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
obs : ∀ᶠ (δ : ℝ) in 𝓝 0, x ∉ thickening δ E
δ : ℝ
x_notin_thE : x ∉ thickening δ E
a✝ : δ ∈ Ioi 0
⊢ (thickening δ E).mulIndicator f x = (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r
ω : α
⊢ f ω ≤ r → f ω ≤ r ∨ f ω ≤ 0

Tactic:
hω

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
r : ℝ
hfint : Integrable f μ
hfint' : 0 ≤ ∫ (x : α), f x ∂μ
hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r
ω : α
hω : f ω ≤ r
⊢ f ω ≤ r ∨ f ω ≤ 0
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
δ : ℝ
⊢ δ ∈ univ →
    (cthickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
_

Goal state after:
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
δ : ℝ
a✝ : δ ∈ univ
⊢ (cthickening δ E).mulIndicator f x =
    (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case neg
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 : α → E
hT : DominatedFinMeasAdditive μ T C
hC : 0 ≤ C
hF : ¬CompleteSpace F
⊢ ‖setToFun μ T hT f‖ ≤ ↑(NNReal.mk C hC) * (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
simp only [setToFun, hF, ↓reduceDIte, norm_zero, NNReal.coe_mk, ofReal_norm]
    positivity

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

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f g : ((i : δ) → X i) → ℝ≥0∞
hst : s ⊆ t
hf : Measurable f
hg : Measurable g
hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ
⊢ ∫⋯∫⁻_t, f ∂μ ≤ ∫⋯∫⁻_t, g ∂μ

Tactic:
rw [← union_sdiff_of_subset hst, lmarginal_union' μ f hf disjoint_sdiff,
    lmarginal_union' μ g hg disjoint_sdiff]
  exact lmarginal_mono hfg

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ leftLim f x = f x

Tactic:
apply le_antisymm (leftLim_le hf (le_refl _))

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
f : α → β
hf : Monotone f
x : α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
h : leftLim f x = rightLim f x
⊢ f x ≤ leftLim f x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ ∫⁻ (a : X) in S \ s i, ‖f a‖ₑ ∂μ ≤ ↑ε

Tactic:
← withDensity_apply₀ _ (hSm.diff (hsm _)),

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
inst✝¹ : Preorder ι
inst✝ : atTop.IsCountablyGenerated
s : ι → Set X
hsm : ∀ (i : ι), NullMeasurableSet (s i) μ
h_mono : Monotone s
hne : atTop.NeBot
this✝ : IsDirectedOrder ι
S : Set X := ⋃ i, s i
hfi : IntegrableOn f S μ
hfi' : (μ.withDensity fun x ↦ ↑‖f x‖₊) S < ∞
hSm : NullMeasurableSet S μ
hsub : ∀ {i : ι}, s i ⊆ S
ν : Measure X := μ.withDensity fun x ↦ ‖f x‖ₑ
hν : ν = μ.withDensity fun x ↦ ‖f x‖ₑ
ε : ℝ≥0
ε0 : 0 < ↑ε
this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
i : ι
hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)
hsm' : NullMeasurableSet (s i) ν
⊢ (μ.withDensity fun a ↦ ‖f a‖ₑ) (S \ s i) ≤ ↑ε
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case e'_3
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := ⋯
⊢ ↑K = ↑(t.sup L)

Tactic:
rw [Compacts.coe_finset_sup, Finset.sup_eq_iSup]

Goal state after:
case e'_3
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
⊢ ↑K = ⨆ a ∈ t, ↑(L a)
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
⊢ i ∈ t → μ (L i) ≤ μ.innerContent (U i)

Tactic:
_

Goal state after:
case convert_2
G : Type w
inst✝¹ : TopologicalSpace G
μ : Content G
inst✝ : R1Space G
U : ℕ → Opens G
h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)
K : Compacts G
hK : ↑K ⊆ ↑(⨆ i, U i)
t : Finset ℕ
ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)
K' : ℕ → Set G
h1K' : ∀ (i : ℕ), IsCompact (K' i)
h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i
h3K' : ↑K = ⋃ i ∈ t, K' i
L : ℕ → Compacts G := fun n ↦ { carrier := K' n, isCompact' := ⋯ }
i : ℕ
a✝ : i ∈ t
⊢ μ (L i) ≤ μ.innerContent (U i)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case neg
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
hts : s ⊆ t
h't : ∀ᵐ (x : X) ∂μ, x ∈ t \ s → f x = 0
h : ¬IntegrableOn f t μ
this : ¬IntegrableOn f s μ
⊢ ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
rw [integral_undef h, integral_undef this]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
⊢ μ K < ∞ → μ.measure ↑K ≤ μ K + ↑ε

Tactic:
_

Goal state after:
case refine_1
G : Type w
inst✝³ : TopologicalSpace G
μ : Content G
inst✝² : MeasurableSpace G
inst✝¹ : R1Space G
inst✝ : BorelSpace G
H : μ.ContentRegular
K : Compacts G
ε : ℝ≥0
εpos : 0 < ε
a✝ : μ K < ∞
⊢ μ.measure ↑K ≤ μ K + ↑ε
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
s : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝ : X
h1 : f a✝ = AEStronglyMeasurable.mk f hfm a✝
⊢ ‖AEStronglyMeasurable.mk f hfm a✝‖ ≤ C → ‖f a✝‖ ≤ C

Tactic:
_

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
s : Set X
μ : Measure X
C : ℝ
hs : μ s < ∞
hC : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖f x‖ ≤ C
hfm : AEStronglyMeasurable f (μ.restrict s)
A : ∀ᵐ (x : X) ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C
B : MeasurableSet {x | ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C}
a✝¹ : X
h1 : f a✝¹ = AEStronglyMeasurable.mk f hfm a✝¹
a✝ : ‖AEStronglyMeasurable.mk f hfm a✝¹‖ ≤ C
⊢ ‖f a✝¹‖ ≤ C