mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: M : Type u_2
ι : Type u_3
α : Type u_4
β : Type u_5
inst✝² : CommMonoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
mβ : MeasurableSpace β
f : ι → α → β → M
g : α → β
s : Finset ι
hf : ∀ i ∈ s, Measurable ↿(f i)
hg : Measurable g
⊢ Measurable fun a ↦ (∏ i ∈ s, f i a) (g a)

Tactic:
simp only [prod_apply]

Goal state after:
M : Type u_2
ι : Type u_3
α : Type u_4
β : Type u_5
inst✝² : CommMonoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
mβ : MeasurableSpace β
f : ι → α → β → M
g : α → β
s : Finset ι
hf : ∀ i ∈ s, Measurable ↿(f i)
hg : Measurable g
⊢ Measurable fun a ↦ ∏ c ∈ s, f c a (g a)
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: ι : Sort u_1
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
f : ι → Set α
⊢ thickening δ (⋃ i, f i) = ⋃ i, thickening δ (f i)

Tactic:
by
  simp_rw [thickening, infEDist_iUnion, iInf_lt_iff, ofPred_exists]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
E : Set α
⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), cthickening ε E

Tactic:
apply cthickening_eq_iInter_cthickening' (Ioi δ) rfl.subset
  simp_rw [inter_eq_right.mpr Ioc_subset_Ioi_self]
  exact fun _ hε => nonempty_Ioc.mpr hε

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

Goal state before:
pp: case neg
α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
s : Set ℝ
hs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty
hs₀ : ¬s ⊆ Ioi 0
δ : ℝ
hδs : δ ∈ s
δ_nonpos : δ ∉ Ioi 0
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E

Tactic:
Set.mem_Ioi,

Goal state after:
case neg
α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
s : Set ℝ
hs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty
hs₀ : ¬s ⊆ Ioi 0
δ : ℝ
hδs : δ ∈ s
δ_nonpos : ¬0 < δ
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case neg
α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
s : Set ℝ
hs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty
hs₀ : ¬s ⊆ Ioi 0
δ : ℝ
hδs : δ ∈ s
δ_nonpos : ¬0 < δ
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E

Tactic:
not_lt

Goal state after:
case neg
α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
s : Set ℝ
hs : ∀ (ε : ℝ), 0 < ε → (s ∩ Ioc 0 ε).Nonempty
hs₀ : ¬s ⊆ Ioi 0
δ : ℝ
hδs : δ ∈ s
δ_nonpos : δ ≤ 0
⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ ∈ s, cthickening δ E
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s t : Set α
y : α
r : ℝ
h : y ∈ t
H : hausdorffDist s t < r
fin : hausdorffEDist s t ≠ ∞
⊢ ∃ x ∈ s, dist x y < r

Tactic:
rw [hausdorffDist_comm] at H
  rw [hausdorffEDist_comm] at fin
  simpa [dist_comm] using exists_dist_lt_of_hausdorffDist_lt h H fin

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α →ₛ ℝ≥0∞
s : Set α
hs : MeasurableSet s
x : α
hx : (f.restrict s) x ≠ 0
hxs : x ∈ s
x✝ : μ (⇑(f.restrict s) ⁻¹' {(f.restrict s) x}) ≠ 0
⊢ (f.restrict s) x ∈ f.range

Tactic:
simp only [f.restrict_apply hs, indicator_of_mem hxs, mem_range_self]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : α →ₛ ℝ≥0∞
s : ι → Set α
hd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s
μ : Measure α
⊢ f.lintegral (μ.restrict (⋃ i, s i)) = ⨆ i, f.lintegral (μ.restrict (s i))

Tactic:
by
  simp only [lintegral, Measure.restrict_iUnion_apply_eq_iSup hd (measurableSet_preimage ..),
    ENNReal.mul_iSup]
  refine finsetSum_iSup fun i j ↦ (hd i j).imp fun k ⟨hik, hjk⟩ ↦ fun a ↦ ?_
  constructor <;> gcongr

Goal state after:
Module: Mathlib.Topology.IndicatorConstPointwise

Goal state before:
pp: α : Type u_1
A : Set α
β : Type u_2
inst✝¹ : Zero β
inst✝ : TopologicalSpace β
ι : Type u_3
L : Filter ι
As : ι → Set α
b : β
nhds_b : {0}ᶜ ∈ 𝓝 b
nhds_o : {b}ᶜ ∈ 𝓝 0
⊢ Tendsto (fun i ↦ (As i).indicator fun x ↦ b) L (𝓝 (A.indicator fun x ↦ b)) ↔
    ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

Tactic:
by
  simp_rw [tendsto_pi_nhds]
  apply forall_congr'
  exact tendsto_indicator_const_apply_iff_eventually' L b nhds_b nhds_o

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

Goal state before:
pp: case neg
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
⊢ Measurable fun x ↦ ⨅ i, f (↑i) x

Tactic:
let reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p

Goal state after:
case neg
α : Type u_1
δ : Type u_4
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
mδ : MeasurableSpace δ
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
ι : Type u_5
ι' : Type u_6
f : ι → δ → α
v : Filter ι
hf : ∀ (i : ι), Measurable (f i)
p : ι' → Prop
s : ι' → Set ι
hv : v.HasCountableBasis p s
hs : ∀ (j : ι'), (s j).Countable
this✝ : Countable (Subtype p)
hp : Nonempty (Subtype p)
H : ¬∃ j, s ↑j = ∅
this : ∀ (i : ι'), Countable ↑(s i)
m : Subtype p → Set δ := fun j ↦ {x | BddBelow (range fun i ↦ f (↑i) x)}
m_meas : ∀ (j : Subtype p), MeasurableSet (m j)
mc_meas : MeasurableSet {x | ∀ (j : Subtype p), x ∉ m j}
j : Subtype p
reparam : δ → Subtype p → Subtype p := fun x ↦ liminfReparam (fun i ↦ f i x) s p
⊢ Measurable fun x ↦ ⨅ i, f (↑i) x
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
⊢ Measurable fun x ↦ ⨆ s, ∑ i ∈ s, f i x

Tactic:
exact .iSup fun s ↦ s.measurable_fun_sum fun i _ => h i

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

Goal state before:
pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n ↦ {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n ↦ sigma_finite_sets n ∩ norm_sets n
h_meas : ∀ (n : ℕ), MeasurableSet (sets n)
h_finite : ∀ (n : ℕ), μ (sets n) < ∞
this : ⋃ i, sigma_finite_sets i ∩ norm_sets i = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i
⊢ univ ∩ univ = univ

Tactic:
Set.inter_univ

Goal state after:
case refine_2
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
norm_sets : ℕ → Set α := fun n ↦ {x | f x ≤ ↑n}
norm_sets_spanning : ⋃ n, norm_sets n = univ
sets : ℕ → Set α := fun n ↦ sigma_finite_sets n ∩ norm_sets n
h_meas : ∀ (n : ℕ), MeasurableSet (sets n)
h_finite : ∀ (n : ℕ), μ (sets n) < ∞
this : ⋃ i, sigma_finite_sets i ∩ norm_sets i = (⋃ i, sigma_finite_sets i) ∩ ⋃ i, norm_sets i
⊢ univ = univ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDense

Goal state before:
pp: X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
⊢ Measurable f

Tactic:
let : PseudoMetricSpace α := TopologicalSpace.pseudoMetrizableSpacePseudoMetric α

Goal state after:
X : Type u_3
Y : Type u_4
α : Type u_5
inst✝¹⁰ : Zero α
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace X
inst✝⁶ : MeasurableSpace Y
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : OpensMeasurableSpace Y
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetrizableSpace α
inst✝¹ : MeasurableSpace α
inst✝ : BorelSpace α
f : X × Y → α
hf : Continuous[instTopologicalSpaceProd, inst✝³] f
h'f : HasCompactSupport f
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
⊢ Measurable f
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
f : α →ₛ ℝ≥0∞
μ : Measure α
⊢ ∫⁻ (a : α), f a ∂μ = f.lintegral μ

Tactic:
rw [MeasureTheory.lintegral]

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

Goal state before:
pp: case h
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 ≠ ∞
h_meas : μ (⇑φ ⁻¹' {∞}) ≠ 0
b : ℝ≥0∞
hb : b < ∞
n : ℕ
hn : b < ↑n * μ (⇑φ ⁻¹' {∞})
x : α
hx : x ∈ ⇑φ ⁻¹' {∞}
⊢ ↑(Function.const α (↑n) x) ≤ φ x

Tactic:
simp only [mem_preimage, mem_singleton_iff] at hx

Goal state after:
case h
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 ≠ ∞
h_meas : μ (⇑φ ⁻¹' {∞}) ≠ 0
b : ℝ≥0∞
hb : b < ∞
n : ℕ
hn : b < ↑n * μ (⇑φ ⁻¹' {∞})
x : α
hx : φ x = ∞
⊢ ↑(Function.const α (↑n) x) ≤ φ x
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 ↦ c / ‖(hf.approx n) x‖) atTop (𝓝 (c / ‖f x‖))

Tactic:
exact Tendsto.div tendsto_const_nhds h_tendsto.norm hfx0

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

Goal state before:

Tactic:
(map (↑) ψ).lintegral (μ.restrict s) ≤
        (map (↑) φ).lintegral (μ.restrict s) + (map (↑) (ψ - φ)).lintegral (μ.restrict s) := by
      rw [← SimpleFunc.add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
      refine SimpleFunc.lintegral_mono (fun x => ?_) le_rfl
      simp only [add_tsub_eq_max, le_max_right, coe_map, Function.comp_apply, SimpleFunc.coe_add,
        SimpleFunc.coe_sub, Pi.add_apply, Pi.sub_apply, ENNReal.coe_max (φ x) (ψ x)]
    _ ≤ (map (↑) φ).lintegral (μ.restrict s) + ε₁ := by
      gcongr
      refine le_trans ?_ (hφ _ hψ).le
      exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self
    _ ≤ (SimpleFunc.const α (C : ℝ≥0∞)).lintegral (μ.restrict s) + ε₁ := by
      gcongr
      exact fun x ↦ ENNReal.coe_le_coe.2 (hC x)
    _ = C * μ s + ε₁ := by
      simp only [← SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,
        Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]
    _ ≤ C * ((ε₂ - ε₁) / C) + ε₁ := by gcongr
    _ ≤ ε₂ - ε₁ + ε₁ := by gcongr; apply mul_div_le
    _ = ε₂ := tsub_add_cancel_of_le hε₁₂.le

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
hf : Continuous[inst✝³, inst✝¹] f
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology α
⊢ StronglyMeasurable f

Tactic:
stronglyMeasurable_iff_measurable_separable

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝⁴ : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
h : SecondCountableTopologyEither α β
f : α → β
hf : Continuous[inst✝³, inst✝¹] f
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology α
⊢ Measurable f ∧ IsSeparable (range f)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : One β
f : α → β
hf : Continuous[inst✝⁴, inst✝²] f
k : Set α
hk : IsCompact k
h'f : mulSupport f ⊆ k
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ StronglyMeasurable f

Tactic:
stronglyMeasurable_iff_measurable_separable

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : One β
f : α → β
hf : Continuous[inst✝⁴, inst✝²] f
k : Set α
hk : IsCompact k
h'f : mulSupport f ⊆ k
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
this : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
⊢ Measurable f ∧ IsSeparable (range f)
Module: Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
⊢ Tendsto (fun i ↦ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑i) atTop
    (𝓝 (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s))

Tactic:
← tsub_zero (⨆ (s) (_ : MeasurableSet s) (_ : SigmaFinite (μ.restrict s)), ν s)

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
⊢ Tendsto (fun i ↦ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑i) atTop
    (𝓝 ((⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 0))
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : AddZeroClass β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
ind : ∀ (c : β) ⦃s : Set α⦄ (hs : MeasurableSet s), P (s.indicator fun x ↦ c) ⋯
add :
  ∀ ⦃f g : α → β⦄ (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hfg : StronglyMeasurable (f + g)),
    Disjoint (support f) (support g) → P f hf → P g hg → P (f + g) hfg
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f : α → β
hf : StronglyMeasurable f
s : ℕ → α →ₛ β := hf.approx
⊢ P f hf

Tactic:
refine lim (fun n ↦ (s n).stronglyMeasurable) hf (fun n ↦ ?_) hf.tendsto_approx

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : AddZeroClass β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
ind : ∀ (c : β) ⦃s : Set α⦄ (hs : MeasurableSet s), P (s.indicator fun x ↦ c) ⋯
add :
  ∀ ⦃f g : α → β⦄ (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hfg : StronglyMeasurable (f + g)),
    Disjoint (support f) (support g) → P f hf → P g hg → P (f + g) hfg
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f : α → β
hf : StronglyMeasurable f
s : ℕ → α →ₛ β := hf.approx
n : ℕ
⊢ P ⇑(s n) ⋯
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
const : ∀ (c : β), P (fun x ↦ c) ⋯
pcw :
  ∀ ⦃f g : α → β⦄ {s : Set α} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hs : MeasurableSet s),
    P f hf → P g hg → P (s.piecewise f g) ⋯
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f : α → β
hf : StronglyMeasurable f
s : ℕ → α →ₛ β := hf.approx
⊢ P f hf

Tactic:
refine lim (fun n ↦ (s n).stronglyMeasurable) hf (fun n ↦ ?_) hf.tendsto_approx

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Nonempty β
inst✝ : TopologicalSpace β
P : (f : α → β) → StronglyMeasurable f → Prop
const : ∀ (c : β), P (fun x ↦ c) ⋯
pcw :
  ∀ ⦃f g : α → β⦄ {s : Set α} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hs : MeasurableSet s),
    P f hf → P g hg → P (s.piecewise f g) ⋯
lim :
  ∀ ⦃f : ℕ → α → β⦄ ⦃g : α → β⦄ (hf : ∀ (n : ℕ), StronglyMeasurable (f n)) (hg : StronglyMeasurable g),
    (∀ (n : ℕ), P (f n) ⋯) → (∀ (x : α), Tendsto (fun x_1 ↦ f x_1 x) atTop (𝓝 (g x))) → P g hg
f : α → β
hf : StronglyMeasurable f
s : ℕ → α →ₛ β := hf.approx
n : ℕ
⊢ P ⇑(s n) ⋯
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : T1Space α
h : Perfect (closure[inst✝¹] C)
x : α
xC : x ∈ C
H : ∃ᶠ (y : α) in 𝓝 x, y ≠ x ∧ y ∈ closure[inst✝¹] C
this : ∀ (y : α), y ≠ x ∧ y ∈ closure[inst✝¹] C → ∃ᶠ (z : α) in 𝓝 y, z ≠ x ∧ z ∈ C
⊢ ∃ᶠ (x' : α) in 𝓝 x, ∃ᶠ (x'' : α) in 𝓝 x', x'' ≠ x ∧ x'' ∈ C

Tactic:
exact H.mono this

Goal state after:
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case right
α : Type u_1
inst✝ : TopologicalSpace α
C U : Set α
hC : Perfect C
x : α
xC : x ∈ C
xU : x ∈ U
Uop : IsOpen[inst✝] U
⊢ (closure[inst✝] (U ∩ C)).Nonempty

Tactic:
apply Nonempty.closure

Goal state after:
case right
α : Type u_1
inst✝ : TopologicalSpace α
C U : Set α
hC : Perfect C
x : α
xC : x ∈ C
xU : x ∈ U
Uop : IsOpen[inst✝] U
⊢ (U ∩ C).Nonempty
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: case zero
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
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 ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
x : α
⊢ g 0 x ≤ g' 0 x

Tactic:
simp [g']

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

Goal state before:
pp: case succ
α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
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 ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
x : α
n✝ : ℕ
⊢ g (n✝ + 1) x ≤ g' (n✝ + 1) x

Tactic:
simp [g']

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
g : ι → ℝ
hg : Summable g
h : ∀ᶠ (i : ι) in cofinite, ‖f i‖ ≤ g i
ε : ℝ
hε : ε > 0
s : Finset ι
hs : ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, g i‖ < ε
⊢ ∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε

Tactic:
classical
  refine ⟨s ∪ h.toFinset, fun t ht => ?_⟩
  have : ∀ i ∈ t, ‖f i‖ ≤ g i := by
    intro i hi
    simp only [disjoint_left, mem_union, not_or, h.mem_toFinset, Set.mem_compl_iff,
      Classical.not_not] at ht
    exact (ht hi).2
  calc
    ‖∑ i ∈ t, f i‖ ≤ ∑ i ∈ t, g i := norm_sum_le_of_le _ this
    _ ≤ ‖∑ i ∈ t, g i‖ := le_abs_self _
    _ < ε := hs _ (ht.mono_right le_sup_left)

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 : 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 ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := ⋯
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ g' i a

Tactic:
exact Eventually.of_forall (fun x ↦ monotone_nat_of_le_succ (fun n ↦ le_max_right _ _))

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

Goal state before:
pp: case mp.succ.inr
α : Type u_2
x y : ℕ → α
n : ℕ
ih : res x n = res y n → ∀ ⦃m : ℕ⦄, m < n → x m = y m
m : ℕ
h : x n = y n ∧ res x n = res y n
hm : m = n
⊢ x m = y m

Tactic:
rw [hm]

Goal state after:
case mp.succ.inr
α : Type u_2
x y : ℕ → α
n : ℕ
ih : res x n = res y n → ∀ ⦃m : ℕ⦄, m < n → x m = y m
m : ℕ
h : x n = y n ∧ res x n = res y n
hm : m = n
⊢ x n = y n
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
x✝ : TopologicalSpace (X ⊕ Y) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
⊢ toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ

Tactic:
funext

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
hΦ : Isometry Φ
hΨ : Isometry Ψ
i : PseudoMetricSpace (X ⊕ Y) := gluePremetric hΦ hΨ
x✝¹ : TopologicalSpace (X ⊕ Y) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace
x✝ : Z
⊢ (toGlueL hΦ hΨ ∘ Φ) x✝ = (toGlueR hΦ hΨ ∘ Ψ) x✝
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: case e'_4
α : Type u_1
inst✝ : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ε = 2 * (ε / 2)

Tactic:
rw [mul_comm, ENNReal.div_mul_cancel] <;> norm_num

Goal state after:
Module: Mathlib.Dynamics.Ergodic.MeasurePreserving

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
hs : NullMeasurableSet s μ
n : ℕ
⊢ μ (s ∆ (f^[n] ⁻¹' s)) ≤ n • μ (s ∆ (f ⁻¹' s))

Tactic:
induction n with
  | zero => simp
  | succ n ih =>
    simp only [add_smul, one_smul]
    grw [← ih, measure_symmDiff_le s (f^[n] ⁻¹' s) (f^[n + 1] ⁻¹' s)]
    replace hs : NullMeasurableSet (s ∆ (f ⁻¹' s)) μ :=
      hs.symmDiff <| hs.preimage hf.quasiMeasurePreserving
    rw [iterate_succ', preimage_comp, ← preimage_symmDiff, (hf.iterate n).measure_preimage hs]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
s : Set β
hf : Measurable f
hs : MeasurableSet s
c : ℝ≥0∞
⊢ c * (Measure.map f μ) s = c * μ (f ⁻¹' s)

Tactic:
Measure.map_apply hf hs

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : α → β
s : Set β
hf : Measurable f
hs : MeasurableSet s
c : ℝ≥0∞
⊢ c * μ (f ⁻¹' s) = c * μ (f ⁻¹' s)