mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
inst✝ : CountablyGenerated α
⊢ Pairwise (Function.onFun Disjoint (countablyGeneratedAtom α))

Tactic:
by
  intro p q hpq s hsp hsq
  simp only [bot_eq_empty, subset_empty_iff] at hsp hsq ⊢
  ext x
  simp only [mem_empty_iff_false, iff_false]
  intro hxs
  obtain ⟨n, hn⟩ : ∃ n, p n ≠ q n := by grind
  specialize hsp hxs
  specialize hsq hxs
  simp only [countablyGeneratedAtom, mem_iInter] at hsp hsq
  grind

Goal state after:
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 g : α → β
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
⊢ s.piecewise f g =ᶠ[ae μ ⊓ 𝓟 sᶜ] g

Tactic:
exact (piecewise_eqOn_compl s f g).eventuallyEq.filter_mono inf_le_right

Goal state after:
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
⊢ s.indicator f =ᵐ[μ.restrict sᶜ] 0

Tactic:
classical exact piecewise_ae_eq_restrict_compl hs

Goal state after:
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
hf : f =ᵐ[μ.restrict sᶜ] 0
⊢ s.indicator f =ᵐ[μ] f

Tactic:
rw [Filter.EventuallyEq, ae_restrict_iff' hs.compl] at hf

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
hf : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → f x = 0 x
⊢ s.indicator f =ᵐ[μ] f
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
x y : α
hxy : mapNatBool α x = mapNatBool α y
⊢ x = y

Tactic:
rw [← generateFrom_natGeneratingSequence α] at *

Goal state after:
α : Type u_1
inst✝³ : MeasurableSpace α
inst✝² : CountablyGenerated α
inst✝¹ : CountablyGenerated α
inst✝ : SeparatesPoints α
x y : α
hxy : mapNatBool α x = mapNatBool α y
⊢ x = y
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α : Type u_1
t : ℕ → Set α
n : ℕ
s : Set α
hs : s ∈ memPartition t n
⊢ MeasurableSet s

Tactic:
by
  rw [← sdiff_union_inter s (t n)]
  refine MeasurableSet.union ?_ ?_ <;>
    · refine measurableSet_generateFrom ?_
      rw [memPartition_succ]
      exact ⟨s, hs, by simp⟩

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
f : α → β
hf : MeasurableEmbedding f
inst✝ : SigmaFinite μ
⊢ SigmaFinite (Measure.map f μ)

Tactic:
by
  refine ⟨fun n ↦ f '' (spanningSets μ n) ∪ (Set.range f)ᶜ, by simp, fun n ↦ ?_, ?_⟩
  · rw [hf.map_apply, Set.preimage_union]
    simp only [Set.preimage_compl, Set.preimage_range, Set.compl_univ, Set.union_empty,
      Set.preimage_image_eq _ hf.injective]
    exact measure_spanningSets_lt_top μ n
  · rw [← Set.iUnion_union, ← Set.image_iUnion, iUnion_spanningSets,
      Set.image_univ, Set.union_compl_self]

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
x : α
hx : ∃ i, ∃ (_ : i ∈ F.powerset ∧ i ∉ {∅}), x ∈ P i
C : Finset ι
left✝ : C ∈ F.powerset
C_nonempty : C ∉ {∅}
hxC : x ∈ P C
⊢ x ∈ ⋃ i, s i

Tactic:
have ⟨i, hi⟩ := Finset.nonempty_iff_ne_empty.mpr <| Finset.notMem_singleton.mp C_nonempty

Goal state after:
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
x : α
hx : ∃ i, ∃ (_ : i ∈ F.powerset ∧ i ∉ {∅}), x ∈ P i
C : Finset ι
left✝ : C ∈ F.powerset
C_nonempty : C ∉ {∅}
hxC : x ∈ P C
i : ι
hi : i ∈ C
⊢ x ∈ ⋃ i, s i
Module: Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated

Goal state before:
pp: α✝ : Type u_1
β : Type u_2
m : MeasurableSpace α✝
h : CountablyGenerated α✝
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
hα : CountableOrCountablyGenerated α γ
hβ : CountableOrCountablyGenerated β γ
⊢ CountableOrCountablyGenerated (α × β) γ

Tactic:
rcases hα with (hα | hα) <;> rcases hβ with (hβ | hβ)

Goal state after:
case inl.inl
α✝ : Type u_1
β : Type u_2
m : MeasurableSpace α✝
h : CountablyGenerated α✝
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
hα : Countable α
hβ : Countable β
⊢ CountableOrCountablyGenerated (α × β) γ

case inl.inr
α✝ : Type u_1
β : Type u_2
m : MeasurableSpace α✝
h : CountablyGenerated α✝
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
hα : Countable α
hβ : CountablyGenerated γ
⊢ CountableOrCountablyGenerated (α × β) γ

case inr.inl
α✝ : Type u_1
β : Type u_2
m : MeasurableSpace α✝
h : CountablyGenerated α✝
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
hα : CountablyGenerated γ
hβ : Countable β
⊢ CountableOrCountablyGenerated (α × β) γ

case inr.inr
α✝ : Type u_1
β : Type u_2
m : MeasurableSpace α✝
h : CountablyGenerated α✝
α : Type u_3
γ : Type u_4
inst✝ : MeasurableSpace γ
hα hβ : CountablyGenerated γ
⊢ CountableOrCountablyGenerated (α × β) γ
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
⊢ ∑ i ∈ F, ∑' (C : ↑(G i)), (μ.restrict (P ↑C)) t =
    ∑' (C : Finset ι), ∑ i ∈ F, (G i).indicator (fun C ↦ (μ.restrict (P C)) t) C

Tactic:
rw [Summable.tsum_finsetSum (fun _ _ ↦ ENNReal.summable)]

Goal state after:
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
⊢ ∑ i ∈ F, ∑' (C : ↑(G i)), (μ.restrict (P ↑C)) t =
    ∑ i ∈ F, ∑' (b : Finset ι), (G i).indicator (fun C ↦ (μ.restrict (P C)) t) b
Module: Mathlib.MeasureTheory.Group.Arithmetic

Goal state before:
pp: α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
⊢ Measurable fun p ↦ p.1 / p.2

Tactic:
simp only [div_eq_mul_inv]

Goal state after:
α : Type u_1
G : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : DivInvMonoid G
inst✝¹ : MeasurableMul₂ G
inst✝ : MeasurableInv G
⊢ Measurable fun p ↦ p.1 * p.2⁻¹
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ MeasurableSet (f ⁻¹' s)ᶜ

Tactic:
← preimage_compl

Goal state after:
γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)
s : Set γ
hs : IsOpen[inst✝³] s
⊢ MeasurableSet (f ⁻¹' sᶜ)
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

Goal state before:
pp: γ : Type u_3
δ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : MeasurableSpace γ
inst✝¹ : BorelSpace γ
inst✝ : MeasurableSpace δ
f : δ → γ
hf : ∀ (s : Set γ), IsClosed[inst✝³] s → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)
⊢ Measurable f

Tactic:
apply measurable_of_isClosed; intro s hs
  rcases eq_empty_or_nonempty s with h1 | h1
  · simp [h1]
  by_cases h2 : s = univ
  · simp [h2]
  exact hf s hs h1 h2

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
a : α
⊢ (𝓝 a).IsMeasurablyGenerated

Tactic:
rw [nhds, iInf_subtype']

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
γ₂ : Type u_4
δ : Type u_5
ι : Sort y
s t u : Set α
inst✝¹² : TopologicalSpace α
inst✝¹¹ : MeasurableSpace α
inst✝¹⁰ : OpensMeasurableSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : OpensMeasurableSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : TopologicalSpace γ₂
inst✝² : MeasurableSpace γ₂
inst✝¹ : BorelSpace γ₂
inst✝ : MeasurableSpace δ
a : α
⊢ (⨅ x, 𝓟 ↑x).IsMeasurablyGenerated
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

Goal state before:
pp: case right
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ ∀ i ∈ u, infEDist i s ≤ hausdorffEDist s t + hausdorffEDist t u

Tactic:
change ∀ x ∈ u, infEDist x s ≤ hausdorffEDist s t + hausdorffEDist t u

Goal state after:
case right
α : Type u
inst✝ : PseudoEMetricSpace α
s t u : Set α
⊢ ∀ x ∈ u, infEDist x s ≤ hausdorffEDist s t + hausdorffEDist t u
Module: Mathlib.Topology.MetricSpace.Thickening

Goal state before:
pp: case inl
α : Type u
inst✝ : PseudoEMetricSpace α
A : Set α
r₁ r₂ : ℝ
hr : r₁ < r₂
h₁ : r₁ ≤ 0
⊢ Disjoint (frontier (thickening r₁ A)) (frontier (thickening r₂ A))

Tactic:
simp [thickening_of_nonpos h₁]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
hs : Bornology.IsBounded s
hy : y ∈ s
⊢ (edist x y).toReal ≤ (infEDist x s).toReal + (ediam s).toReal

Tactic:
exact toReal_le_add (edist_le_infEDist_add_ediam hy) (infEDist_ne_top ⟨y, hy⟩) hs.ediam_ne_top

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_nn : 0 ≤ δ
E : Set α
⊢ cthickening δ E = ⋂ ε, ⋂ (_ : δ < ε), thickening ε E

Tactic:
apply cthickening_eq_iInter_thickening' δ_nn (Ioi δ) rfl.subset

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_nn : 0 ≤ δ
E : Set α
⊢ ∀ (ε : ℝ), δ < ε → (Ioi δ ∩ Ioc δ ε).Nonempty
Module: Mathlib.Topology.MetricSpace.HausdorffDistance

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

Tactic:
have r0 : 0 < r := lt_of_le_of_lt hausdorffDist_nonneg H
  have : hausdorffEDist s t < ENNReal.ofReal r := by
    rwa [hausdorffDist, ← ENNReal.toReal_ofReal (le_of_lt r0),
      ENNReal.toReal_lt_toReal fin ENNReal.ofReal_ne_top] at H
  rcases exists_edist_lt_of_hausdorffEDist_lt h this with ⟨y, hy, yr⟩
  rw [edist_dist, ENNReal.ofReal_lt_ofReal_iff r0] at yr
  exact ⟨y, hy, yr⟩

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
h : ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k < f a
⊢ False

Tactic:
rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f
a : α
h : ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k < f a
q : ℚ
left✝ : 0 ≤ q
lt_q : ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k < ↑(↑q).toNNReal
q_lt : ↑(↑q).toNNReal < f a
⊢ False
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case pos
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
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
a✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' ↦ IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := (aeSeqSet hf p).piecewise g fun x ↦ hα.some
b : δ
h : b ∈ aeSeqSet hf p
⊢ IsLUB {a | ∃ i, mk (f i) ⋯ b = a} (g b)

pp: case neg
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
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
a✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' ↦ IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := (aeSeqSet hf p).piecewise g fun x ↦ hα.some
b : δ
h : b ∉ aeSeqSet hf p
⊢ IsLUB {a | ∃ i, ⋯.some = a} hα.some

Tactic:
· have h_set_eq : { a : α | ∃ i : ι, (hf i).mk (f i) b = a } =
        { a : α | ∃ i : ι, f i b = a } := by
        ext x
        simp_rw [Set.mem_ofPred_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h]
      rw [h_set_eq]
      exact aeSeq.fun_prop_of_mem_aeSeqSet hf h

Goal state after:
case neg
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
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
hg : ∀ᵐ (b : δ) ∂μ, IsLUB {a | ∃ i, f i b = a} (g b)
a✝ : Nontrivial α
hα : Nonempty α
hι : Nonempty ι
p : δ → (ι → α) → Prop := fun x f' ↦ IsLUB {a | ∃ i, f' i = a} (g x)
g_seq : δ → α := (aeSeqSet hf p).piecewise g fun x ↦ hα.some
b : δ
h : b ∉ aeSeqSet hf p
⊢ IsLUB {a | ∃ i, ⋯.some = a} hα.some
Module: Mathlib.MeasureTheory.Constructions.BorelSpace.Order

Goal state before:
pp: case neg
α : Type u_1
inst✝⁴ : TopologicalSpace α
mα : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
H : ¬∃ x ∈ s, IsTop x
⊢ MeasurableSet s

Tactic:
· refine .of_mem_nhdsGT_aux h ?_
    simp only [IsTop] at H
    push Not at H
    exact H

Goal state after:
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 : LinearOrder 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)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
⊢ MeasurableSet {b | BddAbove (range fun i ↦ f i b)}

Tactic:
have : {b | BddAbove (range (fun i ↦ f i b))} = {x | ∃ n, ∀ i, f i x ≤ u n} := by
    apply Subset.antisymm
    · rintro x ⟨c, hc⟩
      obtain ⟨n, hn⟩ : ∃ n, c ≤ u n := (tendsto_atTop.1 hu c).exists
      exact ⟨n, fun i ↦ (hc ((mem_range_self i))).trans hn⟩
    · rintro x ⟨n, hn⟩
      refine ⟨u n, ?_⟩
      rintro - ⟨i, rfl⟩
      exact hn i

Goal state after:
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 : LinearOrder 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)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
B : ∀ (c : α), MeasurableSet {x | ∀ (i : ι), f i x ≤ c}
u : ℕ → α
hu : Tendsto u atTop atTop
this : {b | BddAbove (range fun i ↦ f i b)} = {x | ∃ n, ∀ (i : ι), f i x ≤ u n}
⊢ MeasurableSet {b | BddAbove (range fun i ↦ f i b)}
Module: Mathlib.Topology.Instances.Real.Lemmas

Goal state before:
pp: ⊢ ∀ u ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b}, IsOpen u

Tactic:
simp +contextual [isOpen_Ioo]

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
⊢ (f.restrict s).lintegral μ = ∑ r ∈ f.range, r * μ (⇑f ⁻¹' {r} ∩ s)

Tactic:
classical
  exact calc
    (restrict f s).lintegral μ = ∑ r ∈ f.range, r * μ (restrict f s ⁻¹' {r}) :=
      lintegral_eq_of_subset _ fun x hx =>
        if hxs : x ∈ s then fun _ => by
          simp only [f.restrict_apply hs, indicator_of_mem hxs, mem_range_self]
        else False.elim <| hx <| by simp [*]
    _ = ∑ r ∈ f.range, r * μ (f ⁻¹' {r} ∩ s) :=
      Finset.sum_congr rfl <|
        forall_mem_range.2 fun b =>
          if hb : f b = 0 then by simp only [hb, zero_mul]
          else by rw [restrict_preimage_singleton _ hs hb, inter_comm]

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
inst✝ : MeasurableSpace β
f : α →ₛ ℝ≥0∞
g : β →ₛ ℝ≥0∞
ν : Measure β
H : ∀ (y : ℝ≥0∞), μ (⇑f ⁻¹' {y}) = ν (⇑g ⁻¹' {y})
⊢ ∑ x ∈ f.range, x * μ (⇑f ⁻¹' {x}) = ∑ x ∈ g.range, x * μ (⇑f ⁻¹' {x})

Tactic:
apply lintegral_eq_of_subset

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
f : α →ₛ ℝ≥0∞
g : β →ₛ ℝ≥0∞
ν : Measure β
H : ∀ (y : ℝ≥0∞), μ (⇑f ⁻¹' {y}) = ν (⇑g ⁻¹' {y})
⊢ ∀ (x : α), f x ≠ 0 → μ (⇑f ⁻¹' {f x}) ≠ 0 → f x ∈ g.range
Module: Mathlib.MeasureTheory.Function.SimpleFunc

Goal state before:
pp: case insert
α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddZeroClass γ
motive : (α →ₛ γ) → Prop
const :
  ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f → motive g → motive (f + g)
x : γ
s : Finset γ
hxs : x ∉ s
ih : ∀ (f : α →ₛ γ), range ⇑f \ {0} = ↑s → motive f
f : α →ₛ γ
h : range ⇑f \ {0} = ↑(insert x s)
⊢ motive f

Tactic:
have mx := f.measurableSet_preimage {x}
    let g := SimpleFunc.piecewise (f ⁻¹' {x}) mx 0 f
    have Pg : motive g := by
      apply ih
      simp only [g, SimpleFunc.coe_piecewise, range_piecewise]
      rw [image_compl_preimage, union_sdiff_distrib, sdiff_sdiff_comm, h, Finset.coe_insert,
        insert_sdiff_self_of_notMem, sdiff_eq_empty.mpr, Set.empty_union]
      · rw [Set.image_subset_iff]
        convert! Set.subset_univ _
        exact preimage_const_of_mem (mem_singleton _)
      · rwa [Finset.mem_coe]
    convert! add _ Pg (const x mx)
    · ext1 y
      by_cases hy : y ∈ f ⁻¹' {x}
      · simpa [g, hy]
      · simp [g, hy]
    rw [disjoint_iff_inf_le]
    rintro y
    by_cases hy : y ∈ f ⁻¹' {x} <;> simp [g, hy]

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

Goal state before:
pp: α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ 1 ≤ c / ‖(hf.approx n) x‖

Tactic:
· rwa [one_le_div (lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0))]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEStronglyMeasurable.mk f hf x
x : α
hx : x ∈ s → f x = AEStronglyMeasurable.mk f hf x
⊢ x ∈ s → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
intro hx_mem

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = AEStronglyMeasurable.mk f hf x
x : α
hx : x ∈ s → f x = AEStronglyMeasurable.mk f hf x
hx_mem : x ∈ s
⊢ s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
hx : x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
⊢ x ∈ sᶜ → s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x

Tactic:
intro hx_mem

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
f g : α → β
s : Set α
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
hf : AEStronglyMeasurable f (μ.restrict s)
hg : AEStronglyMeasurable g (μ.restrict sᶜ)
h : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
x : α
hx : x ∈ sᶜ → g x = AEStronglyMeasurable.mk g hg x
hx_mem : x ∈ sᶜ
⊢ s.piecewise f g x = s.piecewise (AEStronglyMeasurable.mk f hf) (AEStronglyMeasurable.mk g hg) x
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
⊢ Monotone (aeSeq hf p)

Tactic:
intro n m hnm x

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
n m : ℕ
hnm : n ≤ m
x : α
⊢ aeSeq hf p n x ≤ aeSeq hf p m x