mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
⊢ Filter.map f (ae (μ.restrict s)) ≤ Filter.map (s.indicator f) (ae μ)

Tactic:
intro t

Goal state after:
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
hs : MeasurableSet s
t : Set β
⊢ t ∈ Filter.map (s.indicator f) (ae μ) → t ∈ Filter.map f (ae (μ.restrict s))
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
⊢ ∀ (s : ℕ → Set α),
    (∀ (n : ℕ), MeasurableSet (s n)) →
      (∀ (n : ℕ), ∃ q, s n = ⋃ p, if q p then countablyGeneratedAtom α p else ∅) →
        ∃ q, ⋃ i, s i = ⋃ p, if q p then countablyGeneratedAtom α p else ∅

Tactic:
intro t

Goal state after:
case refine_4
α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
s : Set α
hs : MeasurableSet s
t : ℕ → Set α
⊢ (∀ (n : ℕ), MeasurableSet (t n)) →
    (∀ (n : ℕ), ∃ q, t n = ⋃ p, if q p then countablyGeneratedAtom α p else ∅) →
      ∃ q, ⋃ i, t i = ⋃ p, if q p then countablyGeneratedAtom α p else ∅
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
i : ι
⊢ (μ i) (s i) = 0

Tactic:
rw [measure_toMeasurable]

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
i : ι
⊢ (μ i) {x | f x ≠ mk f ⋯ x} = 0
Module: Mathlib.MeasureTheory.Measure.AEMeasurable

Goal state before:
pp: case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
⊢ Measurable ((⋃ i, (s i)ᶜ).domRestrict f)

Tactic:
intro t

Goal state after:
case refine_2
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) (s i) = 0
hsm : MeasurableSet (⋂ i, s i)
hs : ∀ (i : ι), ∀ x ∉ s i, f x = mk f ⋯ x
g : α → β := (⋂ i, s i).piecewise (const α default) f
t : Set β
⊢ MeasurableSet t → MeasurableSet ((⋃ i, (s i)ᶜ).domRestrict f ⁻¹' t)
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: β : Type u_2
α : Type u_3
γ : Type u_4
inst✝¹ : MeasurableSpace γ
inst✝ : Nonempty α
h : CountableOrCountablyGenerated (α × β) γ
⊢ CountableOrCountablyGenerated β γ

Tactic:
rcases h.countableOrCountablyGenerated with (h | h)
  · have := countable_right_of_prod_of_nonempty h
    infer_instance
  · infer_instance

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

Goal state before:
pp: α : Type u_2
ι : Type u_6
x✝ : MeasurableSpace α
μ : Measure α
s : ι → Set α
M : ℕ
hs_meas : ∀ (i : ι), MeasurableSet (s i)
hs : ∀ (y : α), {i | y ∈ s i}.encard ≤ ↑M
t : Set α
ht : MeasurableSet t
F : Finset ι
P : Finset ι → Set α := fun C ↦ (⋂ i ∈ C, s i) ∩ ⋂ i ∈ F \ C, (s i)ᶜ
Cs : Finset (Finset ι) := F.powerset \ {∅}
G : ι → Set (Finset ι) := fun i ↦ {C | C ∈ F.powerset ∧ i ∈ C}
P_meas : ∀ (C : Finset ι), MeasurableSet (P C)
P_cover : ∀ {i : ι}, i ∈ F → s i ⊆ ⋃ C ∈ G i, P C
iUnion_P : ⋃ C ∈ Cs, P C ⊆ ⋃ i, s i
P_subset_s : ∀ {i : ι} {C : Finset ι}, i ∈ C → P C ⊆ s i
mem_C : ∀ {i : ι}, i ∈ F → ∀ {C : Finset ι} {x : α}, x ∈ P C → x ∈ s i → i ∈ C
C_subset_C : ∀ {C₁ C₂ : Finset ι}, C₁ ∈ Cs → ∀ {x : α}, x ∈ P C₁ ∩ P C₂ → C₁ ⊆ C₂
⊢ M • (μ.restrict (⋃ C ∈ Cs, P C)) t ≤ (M • μ.restrict (⋃ i, s i)) t

Tactic:
rw [Measure.smul_apply]
      exact nsmul_le_nsmul_right (μ.restrict_mono_set iUnion_P t) M

Goal state after:
Module: Mathlib.Topology.Separation.GDelta

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
K : Set X
hK : IsClosed[inst✝²] K
⊢ y ∈ K → x ∈ K

Tactic:
hyK

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PerfectlyNormalSpace X
x y : X
hxy : x ⤳ y
K : Set X
hK : IsClosed[inst✝²] K
hyK : y ∈ K
⊢ x ∈ K
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case refine_1.inl.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
hcovBy : ¬∃ b, a ⋖ b
t : Set α
hat : t ⊆ Ioi a
htc : t.Countable
htU : ⋃ i ∈ t, Ioi i = ⋃ i ∈ Ioi a, Ioi i
⊢ MeasurableSet (Ioi a)

Tactic:
have : Ioi a = ⋃ b ∈ t, Ici b := by
          refine Subset.antisymm ?_ <| iUnion₂_subset fun b hb ↦ Ici_subset_Ioi.2 (hat hb)
          refine Subset.trans ?_ <| iUnion₂_mono fun _ _ ↦ Ioi_subset_Ici_self
          simpa [CovBy, htU, subset_def] using hcovBy

Goal state after:
case refine_1.inl.inr
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopology α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
this✝ : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)
H : ∀ (a : α), MeasurableSet (Iio a)
a : α
hcovBy : ¬∃ b, a ⋖ b
t : Set α
hat : t ⊆ Ioi a
htc : t.Countable
htU : ⋃ i ∈ t, Ioi i = ⋃ i ∈ Ioi a, Ioi i
this : Ioi a = ⋃ b ∈ t, Ici b
⊢ MeasurableSet (Ioi a)
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
s₁ s₂ t₁ t₂ : Set α
⊢ hausdorffEDist (⋃ b, bif b then s₁ else s₂) (⋃ b, bif b then t₁ else t₂) ≤
    max (hausdorffEDist s₁ t₁) (hausdorffEDist s₂ t₂)

Tactic:
sup_eq_iSup

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
s₁ s₂ t₁ t₂ : Set α
⊢ hausdorffEDist (⋃ b, bif b then s₁ else s₂) (⋃ b, bif b then t₁ else t₂) ≤
    ⨆ b, bif b then hausdorffEDist s₁ t₁ else hausdorffEDist s₂ t₂
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set α
x : α
⊢ infEDist x (cthickening ?m.15 s) + ENNReal.ofReal ?m.15 ≤ infEDist x (thickening δ s) + ENNReal.ofReal δ

Tactic:
by gcongr; exact thickening_subset_cthickening ..

Goal state after:
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inl
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : ε ≤ 0
⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s

pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s

Tactic:
· simp only [thickening_of_nonpos hε, empty_subset]

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
⊢ thickening ε (thickening δ s) ⊆ thickening (ε + δ) s
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inl
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : ε ≤ 0
⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s

pp: case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s

Tactic:
· simp only [thickening_of_nonpos hε, empty_subset]

Goal state after:
case inr
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
⊢ thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
⊢ f.restrict univ = f

Tactic:
simp [restrict]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
f : α →ₛ β
⊢ f.restrict ∅ = 0

Tactic:
simp [restrict]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ⨆ n, (approx ennrealRatEmbed f n) a = f a

Tactic:
iSup_approx_apply ennrealRatEmbed f a hf rfl

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
⊢ ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k = f a
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_5
inst✝ : MulZeroClass β
f g : α →ₛ β
hf : f.FinMeasSupp μ
hg : g.FinMeasSupp μ
⊢ (f * g).FinMeasSupp μ

Tactic:
by
  rw [mul_eq_map₂]
  exact hf.map₂ hg (zero_mul 0)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁶ : MeasurableSpace α
inst✝⁵ : TopologicalSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : MeasurableSpace β
inst✝² : BorelSpace β
ι : Type u_3
f : ι → α → β
g : α → β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
this✝ : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
s : Set β
h1s : IsClosed[inst✝⁵] s
h2s : s.Nonempty
h3s : s ≠ Set.univ
this : Measurable fun x ↦ infNndist (g x) s
x : α
⊢ x ∈ g ⁻¹' s ↔ x ∈ (fun x ↦ infNndist (g x) s) ⁻¹' {0}

Tactic:
simp [← h1s.mem_iff_infDist_zero h2s, ← NNReal.coe_eq_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: a : ℝ
this : IsGLB (range Rat.cast ∩ Ioi a) a
⊢ MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Ioi a, Ioi ↑y)

Tactic:
exact MeasurableSet.biUnion (to_countable _)
    fun b _ => GenerateMeasurable.basic (Ioi (b : ℝ)) (by simp)

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case e_a.inl
α : Type u_5
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : ℝ≥0
ht : 1 < t
A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0)
x : α
H : f x = ∞
⊢ 0 < f x ↔ f x = ∞ ∨ 0 < f x ∧ f x < ∞

Tactic:
simp only [H, or_false, ENNReal.zero_lt_top, not_top_lt, and_false]

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

Goal state before:
pp: case hmono
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
⊢ ∀ ⦃s t : Set (X × Y)⦄,
    s ⊆ t →
      (∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
        ∃ g s_1, MeasurableSet s_1 ∧ s ⊆ s_1 ∧ ∀ x ∈ s_1, dist (f x) (g x) < ε

Tactic:
intro t

Goal state after:
case hmono
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
t : Set (X × Y)
⊢ ∀ ⦃t_1 : Set (X × Y)⦄,
    t ⊆ t_1 →
      (∃ g s, MeasurableSet s ∧ t_1 ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
        ∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: case hunion
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
⊢ ∀ ⦃s t : Set (X × Y)⦄,
    (∃ g s_1, MeasurableSet s_1 ∧ s ⊆ s_1 ∧ ∀ x ∈ s_1, dist (f x) (g x) < ε) →
      (∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
        ∃ g s_1, MeasurableSet s_1 ∧ s ∪ t ⊆ s_1 ∧ ∀ x ∈ s_1, dist (f x) (g x) < ε

Tactic:
intro t

Goal state after:
case hunion
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝⁷ : Zero α
inst✝⁶ : TopologicalSpace X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace X
inst✝³ : MeasurableSpace Y
inst✝² : OpensMeasurableSpace X
inst✝¹ : OpensMeasurableSpace Y
inst✝ : PseudoMetricSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h'f : HasCompactSupport f
ε : ℝ
hε : 0 < ε
K : Set (X × Y)
hK : IsCompact K
t : Set (X × Y)
⊢ ∀ ⦃t_1 : Set (X × Y)⦄,
    (∃ g s, MeasurableSet s ∧ t ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
      (∃ g s, MeasurableSet s ∧ t_1 ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε) →
        ∃ g s, MeasurableSet s ∧ t ∪ t_1 ⊆ s ∧ ∀ x ∈ s, dist (f x) (g x) < ε
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Subsingleton β
f_sf : α →ₛ β := { toFun := f, measurableSet_fiber' := ⋯, finite_range' := ⋯ }
⊢ StronglyMeasurable f

Tactic:
exact ⟨fun _ => f_sf, fun x => tendsto_const_nhds⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: case neg
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : ¬‖f x‖ = 0
this : min 1 (c / ‖f x‖) = 1
⊢ Tendsto (fun n ↦ min 1 (c / ‖(hf.approx n) x‖)) atTop (𝓝 (min 1 (c / ‖f x‖)))

Tactic:
refine Tendsto.min tendsto_const_nhds ?_

Goal state after:
case neg
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
hf : StronglyMeasurable f
c : ℝ
x : α
hfx : ‖f x‖ ≤ c
h_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 (f x))
hfx0 : ¬‖f x‖ = 0
this : min 1 (c / ‖f x‖) = 1
⊢ Tendsto (fun n ↦ c / ‖(hf.approx n) x‖) atTop (𝓝 (c / ‖f x‖))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Markov

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
hf : ∀ a ∈ s, a ∈ t → f a ≤ 1
hf' : ∀ a ∈ s, a ∉ t → f a = 0
a : α
has : a ∈ s
⊢ s.indicator f a ≤ 1

Tactic:
by_cases hat : a ∈ t

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
hf : ∀ a ∈ s, a ∈ t → f a ≤ 1
hf' : ∀ a ∈ s, a ∉ t → f a = 0
a : α
has : a ∈ s
hat : a ∈ t
⊢ s.indicator f a ≤ 1

case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
f : α → ℝ≥0∞
hf : ∀ a ∈ s, a ∈ t → f a ≤ 1
hf' : ∀ a ∈ s, a ∉ t → f a = 0
a : α
has : a ∈ s
hat : a ∉ t
⊢ s.indicator f a ≤ 1
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : Countable β
f : β → α → ℝ≥0∞
hf : ∀ (b : β), AEMeasurable (f b) μ
h_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f
⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ

Tactic:
by
  simp_rw [← iSup_apply]
  let p : α → (β → ENNReal) → Prop := fun x f' => Directed LE.le f'
  have hp : ∀ᵐ x ∂μ, p x fun i => f i x := by
    filter_upwards [] with x i j
    obtain ⟨z, hz₁, hz₂⟩ := h_directed i j
    exact ⟨z, hz₁ x, hz₂ x⟩
  have h_ae_seq_directed : Directed LE.le (aeSeq hf p) := by
    intro b₁ b₂
    obtain ⟨z, hz₁, hz₂⟩ := h_directed b₁ b₂
    refine ⟨z, ?_, ?_⟩ <;>
      · intro x
        by_cases hx : x ∈ aeSeqSet hf p
        · repeat rw [aeSeq.aeSeq_eq_fun_of_mem_aeSeqSet hf hx]
          apply_rules [hz₁, hz₂]
        · simp only [aeSeq, hx, if_false]
          exact le_rfl
  convert! lintegral_iSup_directed_of_measurable (aeSeq.measurable hf p) h_ae_seq_directed using 1
  · simp_rw [← iSup_apply]
    rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm]
  · congr 1
    ext1 b
    rw [lintegral_congr_ae]
    apply EventuallyEq.symm
    exact aeSeq.aeSeq_n_eq_fun_n_ae hf hp _

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (a : α) in sᶜᶜ, f a ∂μ ≤ ∫⁻ (a : α) in sᶜᶜ, g a ∂μ

Tactic:
setLIntegral_compl hms.compl,

Goal state after:
case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in sᶜ, f x ∂μ ≤ ∫⁻ (a : α) in sᶜᶜ, g a ∂μ

case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, f x ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in sᶜ, f x ∂μ ≤ ∫⁻ (a : α) in sᶜᶜ, g a ∂μ

pp: case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, f x ∂μ ≠ ∞

Tactic:
setLIntegral_compl hms.compl,

Goal state after:
case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in sᶜ, f x ∂μ ≤ ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α) in sᶜ, g x ∂μ

case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, g x ∂μ ≠ ∞

case right
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ∞
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, f x ∂μ ≠ ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
⊢ MeasurableSet (μ.sigmaFiniteSetWRT ν)

Tactic:
rw [Measure.sigmaFiniteSetWRT]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
⊢ MeasurableSet
    (if h : ∃ s, MeasurableSet s ∧ SigmaFinite (μ.restrict s) ∧ ∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞ then h.choose else ∅)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s t : Set α
⊢ SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT ν))

Tactic:
rw [Measure.sigmaFiniteSetWRT]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s t : Set α
⊢ SigmaFinite
    (μ.restrict
      (if h : ∃ s, MeasurableSet s ∧ SigmaFinite (μ.restrict s) ∧ ∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞ then h.choose else ∅))
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ
⊢ ∫⁻ (a : α), (const α r * eapprox f n) a ∂μ = r * (eapprox f n).lintegral μ

Tactic:
← SimpleFunc.const_mul_lintegral,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ
⊢ ∫⁻ (a : α), (const α r * eapprox f n) a ∂μ = (const α r * eapprox f n).lintegral μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
⊢ r * s.lintegral μ ≤ ∫⁻ (a : α), r * f a ∂μ

Tactic:
← SimpleFunc.const_mul_lintegral,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
s : α →ₛ ℝ≥0∞
hs : ⇑s ≤ fun a ↦ f a
⊢ (const α r * s).lintegral μ ≤ ∫⁻ (a : α), r * f a ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
this : ∀ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ, s ⊆ t → MeasurableSet t → μ t = ∞
⊢ ∀ (i : Set α), s ⊆ i → MeasurableSet i → μ i = ∞

Tactic:
intro t

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
this : ∀ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ, s ⊆ t → MeasurableSet t → μ t = ∞
t : Set α
⊢ s ⊆ t → MeasurableSet t → μ t = ∞
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
⊢ ∀ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ, s ⊆ t → MeasurableSet t → μ t = ∞

Tactic:
intro t

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
t : Set α
⊢ t ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ → s ⊆ t → MeasurableSet t → μ t = ∞
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : SubtractionMonoid β
inst✝ : ContinuousNeg β
hf : FinStronglyMeasurable f μ
n : ℕ
⊢ μ (support fun x ↦ -(hf.approx n) x) < ∞

Tactic:
Function.support_fun_neg (hf.approx n)

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
μ : Measure α
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : SubtractionMonoid β
inst✝ : ContinuousNeg β
hf : FinStronglyMeasurable f μ
n : ℕ
⊢ μ (support ⇑(hf.approx n)) < ∞
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : T1Space α
U : Set α
hu : U.Nontrivial
x : α
h : (U ∩ ({x} ∩ closure[inst✝¹] (U \ {x}))).Nonempty
hx : x ∈ U
⊢ AccPt x (𝓟 U)

Tactic:
apply Set.Nonempty.right at h
    rw [Set.singleton_inter_nonempty, mem_closure_iff_clusterPt,
      ← accPt_principal_iff_clusterPt] at h
    exact h

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ (CauchySeq fun s ↦ ∑ i ∈ s, f i) ↔ ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

Tactic:
cauchySeq_finset_iff_sum_vanishing,

Goal state after:
ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
⊢ (∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ∑ b ∈ t, f b ∈ e) ↔
    ∀ ε > 0, ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε