mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
⊢ MeasurableSet (Iio a)

Tactic:
convert_to MeasurableSet (⋃ (l ∈ t) (u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u)

Goal state after:
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
⊢ Iio a = ⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u ≤ a), Ico l u

case pos
α : Type u_5
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
hd : Dense s
hbot : ∀ (x : α), IsBot x → x ∈ s
hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s
S : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}
this : MeasurableSpace α := MeasurableSpace.generateFrom S
a : α
t : Set α
hts : t ⊆ s
hc : t.Countable
htd : Dense t
htb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t
ha : ∀ b < a, (Ioo b a).Nonempty
⊢ MeasurableSet (⋃ l ∈ t, ⋃ u ∈ t, ⋃ (_ : l < u), ⋃ (_ : u ≤ a), Ico l u)
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr
δ : ℝ
α : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
E : Set α
hδ : 0 ≤ δ
hne : E.Nonempty
⊢ cthickening δ E = ⋃ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] E, closedBall x δ

Tactic:
← cthickening_closure

Goal state after:
case inr
δ : ℝ
α : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : ProperSpace α
E : Set α
hδ : 0 ≤ δ
hne : E.Nonempty
⊢ cthickening δ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E) =
    ⋃ x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] E, closedBall x δ
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case H1
α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r
H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r
x : α
xs : x ∈ s
y : α
yt : y ∈ t
hy : dist x y ≤ r
⊢ infDist x t ≤ r

Tactic:
exact le_trans (infDist_le_dist_of_mem yt) hy

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

Goal state before:
pp: β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
x : δ
⊢ c < (⨆ i, f i) x ↔ ∃ i, ∃ (_ : i ∈ J), c < f i x

Tactic:
iSup_apply,

Goal state after:
β : Type u_2
δ : Type u_4
inst✝⁶ : TopologicalSpace β
mβ : MeasurableSpace β
inst✝⁵ : BorelSpace β
mδ : MeasurableSpace δ
inst✝⁴ : CompleteLinearOrder β
inst✝³ : OrderTopology β
inst✝² : SecondCountableTopology β
ι : Type u_5
inst✝¹ : TopologicalSpace ι
inst✝ : SeparableSpace ι
f : ι → δ → β
mf : ∀ (t : ι), Measurable (f t)
cf : ∀ (x : δ), LowerSemicontinuous fun x_1 ↦ f x_1 x
c : β
J : Set ι
cJ : J.Countable
dJ : Dense J
x : δ
⊢ c < ⨆ i, f i x ↔ ∃ i, ∃ (_ : i ∈ J), c < f i x
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ici x)
⊢ Measurable f

Tactic:
by
  apply measurable_of_Iio
  simp_rw [← compl_Ici, preimage_compl, MeasurableSet.compl_iff]
  assumption

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
a : α
hb : f a ∈ f.range
⊢ g (f a) * μ (⇑(map g f) ⁻¹' {g (f a)}) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})

Tactic:
rw [map_preimage_singleton, ← f.sum_measure_preimage_singleton, Finset.mul_sum]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
g : β → ℝ≥0∞
f : α →ₛ β
a : α
hb : f a ∈ f.range
⊢ ∑ i ∈ f.range with g i = g (f a), g (f a) * μ (⇑f ⁻¹' {i}) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case inr
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hι : Nonempty ι
A : MeasurableSet {b | BddAbove (range fun i ↦ f i b)}
this : Measurable fun _b ↦ sSup ∅
⊢ Measurable fun b ↦ ⨆ i, f i b

Tactic:
apply Measurable.isLUB_of_mem hf A _ _ this

Goal state after:
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hι : Nonempty ι
A : MeasurableSet {b | BddAbove (range fun i ↦ f i b)}
this : Measurable fun _b ↦ sSup ∅
⊢ ∀ b ∈ {b | BddAbove (range fun i ↦ f i b)}, IsLUB {a | ∃ i, f i b = a} (⨆ i, f i b)

α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hι : Nonempty ι
A : MeasurableSet {b | BddAbove (range fun i ↦ f i b)}
this : Measurable fun _b ↦ sSup ∅
⊢ EqOn (fun b ↦ ⨆ i, f i b) (fun _b ↦ sSup ∅) {b | BddAbove (range fun i ↦ f i b)}ᶜ
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: α : Type u
inst✝ : TopologicalSpace α
f : ℝ → α
c : ℝ
hp : Periodic f c
hc : c ≠ 0
hf : Continuous[_, inst✝] f
⊢ IsCompact (f '' [[0, 0 + c]])

Tactic:
exact isCompact_uIcc.image hf

Goal state after:
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: case neg
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
h : Tendsto (fun i ↦ if p i then a else b) L (if q then F else G)
hq : ¬q
⊢ ∀ᶠ (i : ι) in L, p i ↔ q

Tactic:
simp only [hq, ite_false] at h

Goal state after:
case neg
ι : Type u_3
L : Filter ι
β : Type u_4
p : ι → Prop
inst✝¹ : DecidablePred p
q : Prop
inst✝ : Decidable q
a b : β
F G : Filter β
haG : {a}ᶜ ∈ G
hbF : {b}ᶜ ∈ F
haF : 𝓟 {a} ≤ F
hbG : 𝓟 {b} ≤ G
hq : ¬q
h : Tendsto (fun i ↦ if p i then a else b) L G
⊢ ∀ᶠ (i : ι) in L, p i ↔ q
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : Nonempty γ
P : (α →ₛ γ) → Prop
const : ∀ (c : γ), P (SimpleFunc.const α c)
pcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)
f : α →ₛ γ
⊢ P f

Tactic:
let c : γ := Classical.ofNonempty

Goal state after:
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : Nonempty γ
P : (α →ₛ γ) → Prop
const : ∀ (c : γ), P (SimpleFunc.const α c)
pcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)
f : α →ₛ γ
c : γ := Classical.ofNonempty
⊢ P f
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : PseudoEMetricSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : MeasurableSpace β
f : β → α
hf : Measurable f
s : Set α
y₀ : α
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
n : ℕ
x : β
this✝ : Nonempty ↑s
this : ∀ (n : ℕ), Nat.casesOn n y₀ (Subtype.val ∘ denseSeq ↑s) ∈ s
⊢ (approxOn f hf s y₀ h₀ n) x ∈ s

Tactic:
apply this

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
φ : α →ₛ ℝ≥0∞
hφ : ⇑φ ≤ fun a ↦ f a
⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ

Tactic:
by_cases h : ∀ᵐ a ∂μ, φ a ≠ ∞

Goal state after:
case pos
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
φ : α →ₛ ℝ≥0∞
hφ : ⇑φ ≤ fun a ↦ f a
h : ∀ᵐ (a : α) ∂μ, φ a ≠ ∞
⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ

case neg
α : Type u_1
m : MeasurableSpace α
f : α → ℝ≥0∞
μ : Measure α
φ : α →ₛ ℝ≥0∞
hφ : ⇑φ ≤ fun a ↦ f a
h : ¬∀ᵐ (a : α) ∂μ, φ a ≠ ∞
⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: 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 0)
hfx0 : f x = 0
h_tendsto_norm : Tendsto (fun n ↦ ‖(hf.approx n) x‖) atTop (𝓝 0)
n : ℕ
⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) x‖ ≤ ‖1‖ * ‖(hf.approx n) x‖

Tactic:
gcongr
        rw [norm_one, Real.norm_of_nonneg]
        · exact min_le_left _ _
        · exact le_min zero_le_one (div_nonneg ((norm_nonneg _).trans hfx) (norm_nonneg _))

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Monotone (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ

Tactic:
lintegral_congr_ae (aeSeq.iSup hf hp).symm

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Monotone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_mono : Monotone (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ n, aeSeq hf p n) a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
hf : ∀ (n : ℕ), AEMeasurable (f n) μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
this : Monotone fun n ↦ ∫⁻ (x : α), f n x ∂μ
key : ∫⁻ (x : α), F x ∂μ = ⨆ n, ∫⁻ (x : α), f n x ∂μ
⊢ Tendsto (fun n ↦ ∫⁻ (x : α), f n x ∂μ) atTop (𝓝 (∫⁻ (x : α), F x ∂μ))

Tactic:
by
    rw [key]
    exact tendsto_atTop_iSup this

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
⊢ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ s.indicator f a), g.lintegral μ ≤ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral (μ.restrict s)

Tactic:
apply iSup_le (fun g ↦ (iSup_le (fun hg ↦ ?_)))

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
g : α →ₛ ℝ≥0∞
hg : ⇑g ≤ fun a ↦ s.indicator f a
⊢ g.lintegral μ ≤ ⨆ g, ⨆ (_ : ⇑g ≤ fun a ↦ f a), g.lintegral (μ.restrict s)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Add

Goal state before:
pp: case e'_2
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
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ i, f i) a ∂μ = ∫⁻ (a : α), (⨆ i, aeSeq hf p i) a ∂?m.244

Tactic:
lintegral_congr_ae (aeSeq.iSup hf hp).symm

Goal state after:
case e'_2
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
p : α → (β → ℝ≥0∞) → Prop := fun x f' ↦ Directed LE.le f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
h_ae_seq_directed : Directed LE.le (aeSeq hf p)
⊢ ∫⁻ (a : α), (⨆ n, aeSeq hf p n) a ∂μ = ∫⁻ (a : α), (⨆ i, aeSeq hf p i) a ∂?m.244
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
ι : Type u_6
f : ι → α → M
s : Finset ι
hf : ∀ i ∈ s, StronglyMeasurable (f i)
⊢ StronglyMeasurable fun a ↦ ∏ i ∈ s, f i a

Tactic:
by
  simpa only [← Finset.prod_apply] using s.stronglyMeasurable_prod hf

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

Goal state before:
pp: case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
⊢ Monotone fun n ↦ ⇑(eapprox f n) + ⇑(eapprox g n)

Tactic:
intro i j h a
        dsimp
        gcongr <;> exact monotone_eapprox _ h _

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : Measurable f
hg : Measurable g
⊢ ∫⁻ (a : α), ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a ∂μ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ

Tactic:
by
      rw [lintegral_iSup]
      · congr
        funext n
        rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
        simp only [Pi.add_apply, SimpleFunc.coe_add]
      · fun_prop
      · intro i j h a
        dsimp
        gcongr <;> exact monotone_eapprox _ h _

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

Goal state before:
pp: case hf
α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ
⊢ Measurable ⇑(const α r * eapprox f n)

Tactic:
exact SimpleFunc.measurable _

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

Goal state before:
pp: case refine_2
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
k : ℕ
h : ∀ (b : ℕ), k ≤ b → x b ∈ {x | (fun n ↦ AEMeasurable (F n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x}
n✝ : ℕ
⊢ AEMeasurable (F (x (n✝ + k))) μ

Tactic:
refine (h _ ?_).1

Goal state after:
case refine_2
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
ι : Type u_2
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, AEMeasurable (F n) μ
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ∞
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n ↦ F n a) l (𝓝 (f a))
x : ℕ → ι
xl : Tendsto x atTop l
hxl : ∀ s ∈ l, ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ s
k : ℕ
h : ∀ (b : ℕ), k ≤ b → x b ∈ {x | (fun n ↦ AEMeasurable (F n) μ) x} ∩ {x | (fun n ↦ ∀ᵐ (a : α) ∂μ, F n a ≤ bound a) x}
n✝ : ℕ
⊢ k ≤ n✝ + k
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✝ : SFinite ν
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT ν)ᶜ
hνs : ν s ≠ 0
⊢ μ s = ∞

Tactic:
have ⟨ν', hν', hνν', _⟩ := exists_isFiniteMeasure_absolutelyContinuous ν
  have h : ∃ s : Set α, MeasurableSet s ∧ SigmaFinite (μ.restrict s)
      ∧ (∀ t ⊆ sᶜ, ν t ≠ 0 → μ t = ∞) := by
    refine ⟨μ.sigmaFiniteSetWRT' ν', measurableSet_sigmaFiniteSetWRT',
      sigmaFinite_restrict_sigmaFiniteSetWRT' _ _,
      fun t ht_subset hνt ↦ measure_eq_top_of_subset_compl_sigmaFiniteSetWRT' ht_subset ?_⟩
    exact fun hν't ↦ hνt (hνν' hν't)
  rw [Measure.sigmaFiniteSetWRT, dif_pos h] at hs_subset
  exact h.choose_spec.2.2 s hs_subset hνs

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : SFinite ν
hμν : μ ≪ ν
⊢ μ.restrict (μ.sigmaFiniteSetWRT ν)ᶜ = ∞ • ν.restrict (μ.sigmaFiniteSetWRT ν)ᶜ

Tactic:
ext s
  rw [Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT.compl,
    Measure.smul_apply, smul_eq_mul,
    Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT.compl]
  by_cases hνs : ν (s ∩ (μ.sigmaFiniteSetWRT ν)ᶜ) = 0
  · rw [hνs, mul_zero]
    exact hμν hνs
  · rw [ENNReal.top_mul hνs, measure_eq_top_of_subset_compl_sigmaFiniteSetWRT
      Set.inter_subset_right hνs]

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

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
h_bound_finite : ∀ᵐ (a : α) ∂μ, F a ≠ ∞
h_exists : ∀ᵐ (a : α) ∂μ, ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l) then h.choose else ∞
hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F' a))
⊢ F' =ᵐ[μ] F

Tactic:
have hF'_le : F' ≤ᵐ[μ] F := by
    filter_upwards [h_bound, hF'_tendsto] with a h_le h_tendsto
    exact le_of_tendsto' h_tendsto (fun m ↦ h_le _)

Goal state after:
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
h_bound_finite : ∀ᵐ (a : α) ∂μ, F a ≠ ∞
h_exists : ∀ᵐ (a : α) ∂μ, ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l) then h.choose else ∞
hF'_tendsto : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F' a))
hF'_le : F' ≤ᵐ[μ] F
⊢ F' =ᵐ[μ] F
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ U ∩ C ⊆ U

Tactic:
exact inter_subset_left

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T25Space α
hC : Perfect C
y : α
yC : y ∈ C
x : α
xC : x ∈ C
hxy : x ≠ y
U : Set α
xU : x ∈ U
Uop : IsOpen[inst✝¹] U
V : Set α
yV : y ∈ V
Vop : IsOpen[inst✝¹] V
hUV : Disjoint (closure[inst✝¹] U) (closure[inst✝¹] V)
⊢ V ∩ C ⊆ V

Tactic:
exact inter_subset_left

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Countable
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h✝ : (E ∩ D).Countable
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
h : y ∈ V
⊢ y ∈ E ∩ D ∪ V ∩ C

pp: case neg
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Countable
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h✝ : (E ∩ D).Countable
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
h : y ∉ V
⊢ y ∈ E ∩ D ∪ V ∩ C

Tactic:
· exact mem_union_right _ (mem_inter h yC)

Goal state after:
case neg
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed[inst✝¹] C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Countable
x : α
xD : x ∈ D
E : Set α
xE : E ∈ 𝓝 x
h✝ : (E ∩ D).Countable
U : Set α
hUb : U ∈ b
xU : x ∈ U
hU : U ⊆ E
y : α
yU : y ∈ U
yC : y ∈ C
h : y ∉ V
⊢ y ∈ E ∩ D ∪ V ∩ C
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
ε : ℝ
εpos : ε > 0
t : Finset α
ht : ∑' (a : { x // x ∉ t }), u ↑a < ε
x : β
hx : x ∈ s
A : Summable fun n ↦ ‖f n x‖
⊢ ‖∑' (x_1 : { x // x ∉ t }), f (↑x_1) x‖ < ε

Tactic:
apply lt_of_le_of_lt _ ht

Goal state after:
α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
ε : ℝ
εpos : ε > 0
t : Finset α
ht : ∑' (a : { x // x ∉ t }), u ↑a < ε
x : β
hx : x ∈ s
A : Summable fun n ↦ ‖f n x‖
⊢ ‖∑' (x_1 : { x // x ∉ t }), f (↑x_1) x‖ ≤ ∑' (a : { x // x ∉ t }), u ↑a