mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.OuterMeasure.Caratheodory

Goal state before:
pp: case e'_4
α : Type u
m : OuterMeasure α
s : ℕ → Set α
h : ∀ (i : ℕ), m.IsCaratheodory (s i)
hd : Pairwise (Disjoint on s)
t : Set α
⊢ ⨆ n, m (t ∩ ⋃ i, ⋃ (_ : i < n), s i) = ∑' (i : ℕ), m (t ∩ s i)

Tactic:
simp [ENNReal.tsum_eq_iSup_nat, isCaratheodory_sum m h hd]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f g : α → β
h : f =ᵐ[μ] g
hf : ¬AEMeasurable f μ
⊢ ¬AEMeasurable g μ

Tactic:
simpa [← aemeasurable_congr h] using hf

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

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → α
hf : map f μ = μ
s : Set α
hs : NullMeasurableSet s μ
hfm : ¬AEMeasurable f μ
⊢ μ (f ⁻¹' s) = μ s

Tactic:
rw [map_of_not_aemeasurable hfm] at hf
    simp [← hf]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
t s : Set α
hm : NullMeasurableSet t μ
h_fin : μ t ≠ ∞
⊢ μ (t ∪ s) = μ (s ∪ t)

Tactic:
union_comm

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
t s : Set α
hm : NullMeasurableSet t μ
h_fin : μ t ≠ ∞
⊢ μ (s ∪ t) = μ (s ∪ t)
Module: Mathlib.Topology.Order.AtTopBotIxx

Goal state before:
pp: case inr.left
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxL : x_L < y
hxR : y < x_R
⊢ a_L ≤ n

Tactic:
contrapose! hxL

Goal state after:
case inr.left
X : Type u_1
inst✝⁶ : LinearOrder X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : OrderTopology X
α : Type u_2
inst✝³ : LinearOrder α
inst✝² : LocallyFiniteOrder α
inst✝¹ : NoMaxOrder X
inst✝ : NoMinOrder X
f g : α → X
hl : Tendsto f atTop atTop
hu : Tendsto g atBot atBot
x : X
h✝ : Nonempty α
x_L : X
hx_L : x_L < x
x_R : X
hx_R : x < x_R
a_L : α
ha_L : ∀ a ≤ a_L, g a ≤ x_L
a_R : α
ha_R : ∀ a ≥ a_R, x_R ≤ f a
n : α
y : X
hf : f n ≤ y
hg : y ≤ g n
hxR : y < x_R
hxL : n < a_L
⊢ y ≤ x_L
Module: Mathlib.MeasureTheory.Measure.MeasureSpace

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

Tactic:
by
  -- WLOG, `ι = ℕ`
  rcases Countable.exists_injective_nat ι with ⟨e, he⟩
  generalize ht : Function.extend e s ⊥ = t
  replace hd : Directed (· ⊆ ·) t := ht ▸ hd.extend_bot he
  suffices μ (⋃ n, t n) = ⨆ n, μ (t n) by
    simp only [← ht, Function.apply_extend μ, ← iSup_eq_iUnion, iSup_extend_bot he,
      Function.comp_def, Pi.bot_apply, bot_eq_empty, measure_empty] at this
    exact this.trans (iSup_extend_bot he _)
  clear! ι
  -- The `≥` inequality is trivial
  refine le_antisymm ?_ (iSup_le fun i ↦ measure_mono <| subset_iUnion _ _)
  -- Choose `T n ⊇ t n` of the same measure, put `Td n = disjointed T`
  set T : ℕ → Set α := fun n => toMeasurable μ (t n)
  set Td : ℕ → Set α := disjointed T
  have hm : ∀ n, MeasurableSet (Td n) := .disjointed fun n ↦ measurableSet_toMeasurable _ _
  calc
    μ (⋃ n, t n) = μ (⋃ n, Td n) := by rw [iUnion_disjointed, measure_iUnion_toMeasurable]
    _ ≤ ∑' n, μ (Td n) := measure_iUnion_le _
    _ = ⨆ I : Finset ℕ, ∑ n ∈ I, μ (Td n) := ENNReal.tsum_eq_iSup_sum
    _ ≤ ⨆ n, μ (t n) := iSup_le fun I => by
      rcases hd.finset_le I with ⟨N, hN⟩
      calc
        (∑ n ∈ I, μ (Td n)) = μ (⋃ n ∈ I, Td n) :=
          (measure_biUnion_finset ((disjoint_disjointed T).set_pairwise I) fun n _ => hm n).symm
        _ ≤ μ (⋃ n ∈ I, T n) := measure_mono (iUnion₂_mono fun n _hn => disjointed_subset _ _)
        _ = μ (⋃ n ∈ I, t n) := measure_biUnion_toMeasurable I.countable_toSet _
        _ ≤ μ (t N) := measure_mono (iUnion₂_subset hN)
        _ ≤ ⨆ n, μ (t n) := le_iSup (μ ∘ t) N

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
β : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : T1Space β
inst✝¹ : SecondCountableTopology β
inst✝ : Nonempty β
f : α → β
h : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b
⊢ ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)

Tactic:
by
  -- We use an `OuterMeasure` so that the proof works without `Measurable f`
  set m : OuterMeasure β := OuterMeasure.map f μ.toOuterMeasure
  replace h : ∀ b : β, m {b}ᶜ ≠ 0 := fun b => not_eventually.mpr (h b)
  inhabit β
  have : m univ ≠ 0 := ne_bot_of_le_ne_bot (h default) (measure_mono <| subset_univ _)
  rcases exists_mem_forall_mem_nhdsWithin_pos_measure this with ⟨b, -, hb⟩
  simp only [nhdsWithin_univ] at hb
  rcases exists_mem_forall_mem_nhdsWithin_pos_measure (h b) with ⟨a, hab : a ≠ b, ha⟩
  simp only [isOpen_compl_singleton.nhdsWithin_eq hab] at ha
  exact ⟨a, b, hab, ha, hb⟩

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Finite

Goal state before:
pp: α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
m : MeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
h✝ : Nonempty α
inhabited_h : Inhabited α
S : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}
T : Set (Set α)
T_count : T.Countable
TS : T ⊆ S
hT : ⋃₀ T = univ
T_ne : T.Nonempty
f : ℕ → Set α
hf : T = range f
fS : ∀ (n : ℕ), f n ∈ S
x : α
⊢ ∃ t ∈ range f, x ∈ t

Tactic:
have : x ∈ ⋃₀ T := by simp only [hT, mem_univ]
    simpa only [mem_sUnion, exists_prop, ← hf]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
⊢ μ.trim hm = Measure.map id μ

Tactic:
refine @Measure.ext α m _ _ (fun s hs ↦ ?_)

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
s : Set α
hs : MeasurableSet s
⊢ (μ.trim hm) s = (Measure.map id μ) s
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case refine_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : Group β
f g : α → β
h : f / g =ᵐ[μ] 1
x : α
hx : (f / g) x = 1 x
⊢ f x = g x

Tactic:
Pi.div_apply,

Goal state after:
case refine_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : Group β
f g : α → β
h : f / g =ᵐ[μ] 1
x : α
hx : f x / g x = 1 x
⊢ f x = g x
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : Group β
f g : α → β
h : f =ᵐ[μ] g
x : α
hx : f x = g x
⊢ (f / g) x = 1 x

Tactic:
Pi.div_apply,

Goal state after:
case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : Group β
f g : α → β
h : f =ᵐ[μ] g
x : α
hx : f x = g x
⊢ f x / g x = 1 x
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case refine_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝² : Group β
inst✝¹ : LE β
inst✝ : MulRightMono β
f g : α → β
h : 1 ≤ᵐ[μ] g / f
a : α
ha : 1 ≤ (g / f) a
⊢ f a ≤ g a

Tactic:
Pi.div_apply,

Goal state after:
case refine_1
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝² : Group β
inst✝¹ : LE β
inst✝ : MulRightMono β
f g : α → β
h : 1 ≤ᵐ[μ] g / f
a : α
ha : 1 ≤ g a / f a
⊢ f a ≤ g a
Module: Mathlib.MeasureTheory.Measure.Restrict

Goal state before:
pp: case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝² : Group β
inst✝¹ : LE β
inst✝ : MulRightMono β
f g : α → β
h : f ≤ᵐ[μ] g
a : α
ha : f a ≤ g a
⊢ 1 ≤ (g / f) a

Tactic:
Pi.div_apply,

Goal state after:
case refine_2
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝² : Group β
inst✝¹ : LE β
inst✝ : MulRightMono β
f g : α → β
h : f ≤ᵐ[μ] g
a : α
ha : f a ≤ g a
⊢ 1 ≤ g a / f a
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : Countable α
inst✝ : SeparatesPoints α
x : α
⊢ MeasurableSet {x}

Tactic:
choose s hsm hxs hys using fun y (h : x ≠ y) ↦ exists_measurableSet_of_ne h
    convert! MeasurableSet.iInter fun y ↦ .iInter fun h ↦ hsm y h
    ext y
    rcases eq_or_ne x y with rfl | h
    · simpa
    · simp only [mem_singleton_iff, h.symm, false_iff, mem_iInter, not_forall]
      exact ⟨y, h, hys y h⟩

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

Goal state before:
pp: case refine_2
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
μ : Measure α
f : α → β
a✝ : Nontrivial β
inhabited_h : Inhabited β
S : Set (Set β)
hSc : S.Countable
hc : MeasurableSpace.CountablyGenerated β
h : NullMeasurable f μ
T : Set β → Set α
hTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s
hTm : ∀ s ∈ S, MeasurableSet (T s)
hTeq : ∀ s ∈ S, T s =ᵐ[μ] f ⁻¹' s
U : Set β → Set α
hUf : ∀ s ∈ S, U s ⊇ f ⁻¹' s
hUm : ∀ s ∈ S, MeasurableSet (U s)
hUeq : ∀ s ∈ S, U s =ᵐ[μ] f ⁻¹' s
v : Set α := ⋃ s ∈ S, U s \ T s
hvm : MeasurableSet v
hvμ : μ v = 0
⊢ Measurable (vᶜ.domRestrict (v.piecewise (fun x ↦ default) f))

Tactic:
· rw [domRestrict_piecewise_compl, domRestrict_eq]
    refine measurable_generateFrom fun s hs ↦ .of_subtype_image ?_
    rw [preimage_comp, Subtype.image_preimage_coe]
    convert! (hTm s hs).diff hvm using 1
    rw [inter_comm]
    refine Set.ext fun x ↦ and_congr_left fun hxv ↦ ⟨fun hx ↦ ?_, fun hx ↦ hTf s hs hx⟩
    exact by_contra fun hx' ↦ hxv <| mem_biUnion hs ⟨hUf s hs hx, hx'⟩

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

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsClosed s
this✝ : UniformSpace X := pseudoMetrizableSpaceUniformity X
this : (uniformity X).IsCountablyGenerated
U : ℕ → SetRel X X
hUo : ∀ (i : ℕ), U i ∈ uniformity X ∧ IsOpen (U i)
hU : (uniformity X).HasBasis (fun x ↦ True) fun i ↦ id (U i)
⊢ IsGδ (closure s)

Tactic:
← hU.biInter_biUnion_ball

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsClosed s
this✝ : UniformSpace X := pseudoMetrizableSpaceUniformity X
this : (uniformity X).IsCountablyGenerated
U : ℕ → SetRel X X
hUo : ∀ (i : ℕ), U i ∈ uniformity X ∧ IsOpen (U i)
hU : (uniformity X).HasBasis (fun x ↦ True) fun i ↦ id (U i)
⊢ IsGδ (⋂ i, ⋂ (_ : True), ⋃ x ∈ s, UniformSpace.ball x (id (U i)))
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
s t : Set α
ε : ℝ≥0
εpos : 0 < ε
h : infEDist x s + hausdorffEDist s t < ∞
ε0 : ↑ε / 2 ≠ 0
this : infEDist x s < infEDist x s + ↑ε / 2
y : α
ys : y ∈ s
dxy : edist x y < infEDist x s + ↑ε / 2
⊢ infEDist x t ≤ infEDist x s + hausdorffEDist s t + ↑ε

Tactic:
have : hausdorffEDist s t < hausdorffEDist s t + ε / 2 :=
      ENNReal.lt_add_right (ENNReal.add_lt_top.1 h).2.ne ε0

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x : α
s t : Set α
ε : ℝ≥0
εpos : 0 < ε
h : infEDist x s + hausdorffEDist s t < ∞
ε0 : ↑ε / 2 ≠ 0
this✝ : infEDist x s < infEDist x s + ↑ε / 2
y : α
ys : y ∈ s
dxy : edist x y < infEDist x s + ↑ε / 2
this : hausdorffEDist s t < hausdorffEDist s t + ↑ε / 2
⊢ infEDist x t ≤ infEDist x s + hausdorffEDist s t + ↑ε
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inr.inr
α : Type u
inst✝ : PseudoEMetricSpace α
ε δ : ℝ
s : Set α
hε : 0 ≤ ε
hδ : 0 ≤ δ
x : α
⊢ (∃ z, (∃ z_1 ∈ s, edist z z_1 < ENNReal.ofReal δ) ∧ edist x z < ENNReal.ofReal ε) →
    ∃ z ∈ s, edist x z < ENNReal.ofReal ε + ENNReal.ofReal δ

Tactic:
exact fun ⟨y, ⟨z, hz, hy⟩, hx⟩ =>
    ⟨z, hz, (edist_triangle _ _ _).trans_lt <| ENNReal.add_lt_add hx hy⟩

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

Goal state before:
pp: case convert_4
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' Iio x)
⊢ ∀ t ∈ range Iio, MeasurableSet (f ⁻¹' t)

Tactic:
rintro _ ⟨x, rfl⟩; exact hf x

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
f : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ (range fun a ↦ (g (f a)) a) ⊆ ⋃ i ∈ range ⇑f, range ⇑(g i)

Tactic:
by
      rintro _ ⟨a, rfl⟩; simp

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ hausdorffDist {x} {y} = dist x y

Tactic:
rw [hausdorffDist, hausdorffEDist_singleton, dist_edist]

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

Goal state before:
pp: case convert_4
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' Ioi x)
⊢ ∀ t ∈ range Ioi, MeasurableSet (f ⁻¹' t)

Tactic:
rintro _ ⟨x, rfl⟩; exact hf x

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : MeasurableSpace α
f : α →ₛ β
g : α →ₛ γ
b : β
c : γ
⊢ ⇑(f.pair g) ⁻¹' {(b, c)} = ⇑f ⁻¹' {b} ∩ ⇑g ⁻¹' {c}

Tactic:
by
  rw [← singleton_prod_singleton]
  exact pair_preimage _ _ _ _

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

Goal state before:
pp: α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
⊢ Measurable g

Tactic:
rw [‹BorelSpace α›.measurable_eq, borel_eq_generateFrom_Ioi α]

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 : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), Measurable (f i)
hg : ∀ (b : δ), IsLUB (range fun i ↦ f i b) (g b)
⊢ Measurable g
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Real

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ι : Type u_5
f : ι → α → ℝ≥0∞
g : α → ℝ≥0∞
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (i : ι), Measurable (f i)
lim : Tendsto f u (𝓝 g)
⊢ Measurable g

Tactic:
by
  rcases u.exists_seq_tendsto with ⟨x, hx⟩
  rw [tendsto_pi_nhds] at lim
  have : (fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop) = g := by
    ext1 y
    exact ((lim y).comp hx).liminf_eq
  rw [← this]
  change Measurable fun y => liminf (fun n => (f (x n) y : ℝ≥0∞)) atTop
  exact .liminf fun n => hf (x n)

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

Goal state before:
pp: case refine_5
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α✝
mα✝ : MeasurableSpace α✝
α : Type u_5
β : Type u_6
γ : Type u_7
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
⊢ Measurable fun r ↦ (↑r, ⊤).1 * (↑r, ⊤).2

Tactic:
simp_rw [mul_comm _ ⊤]

Goal state after:
case refine_5
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α✝
mα✝ : MeasurableSpace α✝
α : Type u_5
β : Type u_6
γ : Type u_7
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
⊢ Measurable fun r ↦ ⊤ * ↑r
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: 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
⊢ μ s = μ (s ∩ f ⁻¹' Ici ⊥)

Tactic:
Ici_bot,

Goal state after:
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
⊢ μ s = μ (s ∩ f ⁻¹' univ)
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
⊢ ⨆ g, ⨆ (_ : Measurable g), ⨆ (_ : g ≤ f), ∫⁻ (a : α), g a ∂μ = ∫⁻ (a : α), f a ∂μ

Tactic:
apply le_antisymm
  · exact iSup_le fun i => iSup_le fun _ => iSup_le fun h'i => lintegral_mono h'i
  · rw [lintegral]
    refine iSup₂_le fun i hi => le_iSup₂_of_le i i.measurable <| le_iSup_of_le hi ?_
    exact le_of_eq (i.lintegral_eq_lintegral _).symm

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

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this✝ : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
n : ℕ
y : β
hy : y ∈ ↑({y ∈ (fs n).range | y ≠ 0})
this : (y : β) → Decidable (y = 0) := fun y ↦ Classical.propDecidable (y = 0)
⊢ y ≠ 0

Tactic:
rw [Finset.mem_coe, Finset.mem_filter] at hy

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
f : α → β
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
m : MeasurableSpace α
μ : Measure α
hf_meas : StronglyMeasurable f
t : Set α
ht : MeasurableSet t
hft_zero : ∀ x ∈ tᶜ, f x = 0
htμ this✝ : SigmaFinite (μ.restrict t)
S : ℕ → Set α := spanningSets (μ.restrict t)
hS_meas : ∀ (n : ℕ), MeasurableSet (S n)
f_approx : ℕ → α →ₛ β := hf_meas.approx
fs : ℕ → α →ₛ β := fun n ↦ (f_approx n).restrict (S n ∩ t)
h_fs_t_compl : ∀ (n : ℕ), ∀ x ∉ t, (fs n) x = 0
n : ℕ
y : β
hy : y ∈ (fs n).range ∧ y ≠ 0
this : (y : β) → Decidable (y = 0) := fun y ↦ Classical.propDecidable (y = 0)
⊢ y ≠ 0
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Sub

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
h_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
h_anti : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
p : α → (ℕ → ℝ≥0∞) → Prop := fun x f' ↦ Antitone f'
hp : ∀ᵐ (x : α) ∂μ, p x fun i ↦ f i x
n m : ℕ
hnm : n ≤ m
x : α
hx : x ∈ aeSeqSet h_meas p
⊢ aeSeq h_meas p m x ≤ aeSeq h_meas p n x

Tactic:
exact aeSeq.prop_of_mem_aeSeqSet h_meas hx hnm

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
hsm : MeasurableSet s
hfs : ∫⁻ (x : α) in s, f x ∂μ ≠ ∞
⊢ ∫⁻ (x : α) in sᶜ, f x ∂μ = ∫⁻ (x : α), f x ∂μ - ∫⁻ (x : α) in s, f x ∂μ

Tactic:
by
  rw [← lintegral_add_compl (μ := μ) f hsm, ENNReal.add_sub_cancel_left hfs]

Goal state after:
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 : MeasurableSet s
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
hsσ : SigmaFinite (μ.restrict s)
⊢ ν (μ.sigmaFiniteSetWRT' ν) < ν (μ.sigmaFiniteSetWRT' ν) + ν s

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs : MeasurableSet s
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
hsσ : SigmaFinite (μ.restrict s)
⊢ Disjoint (μ.sigmaFiniteSetWRT' ν) s

Tactic:
· exact ENNReal.lt_add_right (measure_ne_top _ _) hνs

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
s : Set α
inst✝ : IsFiniteMeasure ν
hs : MeasurableSet s
hs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ
hνs : ν s ≠ 0
hsσ : SigmaFinite (μ.restrict s)
⊢ Disjoint (μ.sigmaFiniteSetWRT' ν) s
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case pos
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
x : α
hx : x ∈ closure[inst✝] C
h : x ∈ C
⊢ AccPt x (𝓟 (closure[inst✝] C))

Tactic:
apply AccPt.mono _ (principal_mono.mpr subset_closure)

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
x : α
hx : x ∈ closure[inst✝] C
h : x ∈ C
⊢ AccPt x (𝓟 C)
Module: Mathlib.Topology.Perfect

Goal state before:
pp: case neg
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
x : α
hx : x ∈ closure[inst✝] C
h : x ∉ C
⊢ AccPt x (𝓟 (closure[inst✝] C))

Tactic:
apply AccPt.mono _ (principal_mono.mpr subset_closure)

Goal state after:
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
hC : Preperfect C
x : α
hx : x ∈ closure[inst✝] C
h : x ∉ C
⊢ AccPt x (𝓟 C)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
⊢ Measurable (uncurry u)

Tactic:
have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd

Goal state after:
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
⊢ Measurable (uncurry u)
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
⊢ StronglyMeasurable (uncurry u)

Tactic:
have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
    rw [tendsto_pi_nhds]
    exact fun p => ht_sf p.fst p.snd

Goal state after:
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
⊢ StronglyMeasurable (uncurry u)