l3lab/ntp-mathlib huggingface dataset:
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
⊢ Set.Countable (s \ interior s)

Next tactic:
have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      exact False.elim (hx'.2 this)
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x ∈ interior s :=
        mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
      exact False.elim (hx.2 this)
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
⊢ PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x)

Next tactic:
intro x hx x' hx' hxx'
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'

Next tactic:
rcases lt_or_gt_of_ne hxx' with (h' | h')
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · 

Proof state:
case inl
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'

Next tactic:
refine disjoint_left.2 fun z hz h'z => ?_
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      

Proof state:
case inl
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
⊢ False

Next tactic:
have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      

Proof state:
case inl
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x < x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
this : x' ∈ interior s
⊢ False

Next tactic:
exact False.elim (hx'.2 this)
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      exact False.elim (hx'.2 this)
    · 

Proof state:
case inr
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
⊢ (Disjoint on fun x => Ioo x (y x)) x x'

Next tactic:
refine disjoint_left.2 fun z hz h'z => ?_
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      exact False.elim (hx'.2 this)
    · refine disjoint_left.2 fun z hz h'z => ?_
      

Proof state:
case inr
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
⊢ False

Next tactic:
have : x ∈ interior s :=
        mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      exact False.elim (hx'.2 this)
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x ∈ interior s :=
        mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
      

Proof state:
case inr
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
x : α
hx : x ∈ s \ interior s
x' : α
hx' : x' ∈ s \ interior s
hxx' : x ≠ x'
h' : x > x'
z : α
hz : z ∈ (fun x => Ioo x (y x)) x
h'z : z ∈ (fun x => Ioo x (y x)) x'
this : x ∈ interior s
⊢ False

Next tactic:
exact False.elim (hx.2 this)
theorem measurableSet_of_mem_nhdsWithin_Ioi_aux {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x)
    (h' : ∀ x ∈ s, ∃ y, x < y) : MeasurableSet s := by
  choose! M hM using h'
  suffices H : (s \ interior s).Countable by
    have : s = interior s ∪ s \ interior s := by rw [union_diff_cancel interior_subset]
    rw [this]
    exact isOpen_interior.measurableSet.union H.measurableSet
  have A : ∀ x ∈ s, ∃ y ∈ Ioi x, Ioo x y ⊆ s := fun x hx =>
    (mem_nhdsWithin_Ioi_iff_exists_Ioo_subset' (hM x hx)).1 (h x hx)
  choose! y hy h'y using A
  have B : Set.PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x) := by
    intro x hx x' hx' hxx'
    rcases lt_or_gt_of_ne hxx' with (h' | h')
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x' ∈ interior s :=
        mem_interior.2 ⟨Ioo x (y x), h'y _ hx.1, isOpen_Ioo, ⟨h', h'z.1.trans hz.2⟩⟩
      exact False.elim (hx'.2 this)
    · refine disjoint_left.2 fun z hz h'z => ?_
      have : x ∈ interior s :=
        mem_interior.2 ⟨Ioo x' (y x'), h'y _ hx'.1, isOpen_Ioo, ⟨h', hz.1.trans h'z.2⟩⟩
      exact False.elim (hx.2 this)
  

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
M : α → α
hM : ∀ x ∈ s, x < M x
y : α → α
hy : ∀ x ∈ s, y x ∈ Ioi x
h'y : ∀ x ∈ s, Ioo x (y x) ⊆ s
B : PairwiseDisjoint (s \ interior s) fun x => Ioo x (y x)
⊢ Set.Countable (s \ interior s)

Next tactic:
exact B.countable_of_Ioo fun x hx => hy x hx.1
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
⊢ MeasurableSet s

Next tactic:
by_cases H : ∃ x ∈ s, IsTop x
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · 

Proof state:
case pos
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
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

Next tactic:
rcases H with ⟨x₀, x₀s, h₀⟩
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    

Proof state:
case pos.intro.intro
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
⊢ MeasurableSet s

Next tactic:
have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = {x₀} ∪ s \ {x₀} := by 

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
⊢ s = {x₀} ∪ s \ {x₀}

Next tactic:
rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
    

Proof state:
case pos.intro.intro
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet s

Next tactic:
rw [this]
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
    rw [this]
    

Proof state:
case pos.intro.intro
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet ({x₀} ∪ s \ {x₀})

Next tactic:
refine' (measurableSet_singleton _).union _
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
    rw [this]
    refine' (measurableSet_singleton _).union _
    

Proof state:
case pos.intro.intro
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
⊢ MeasurableSet (s \ {x₀})

Next tactic:
have A : ∀ x ∈ s \ {x₀}, x < x₀ := fun x hx => lt_of_le_of_ne (h₀ _) (by simpa using hx.2)
/-- If a set is a right-neighborhood of all of its points, then it is measurable. -/
theorem measurableSet_of_mem_nhdsWithin_Ioi {s : Set α} (h : ∀ x ∈ s, s ∈ 𝓝[>] x) :
    MeasurableSet s := by
  by_cases H : ∃ x ∈ s, IsTop x
  · rcases H with ⟨x₀, x₀s, h₀⟩
    have : s = {x₀} ∪ s \ {x₀} := by rw [union_diff_cancel (singleton_subset_iff.2 x₀s)]
    rw [this]
    refine' (measurableSet_singleton _).union _
    have A : ∀ x ∈ s \ {x₀}, x < x₀ := fun x hx => lt_of_le_of_ne (h₀ _) (by 

Proof state:
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 : BorelSpace α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : MeasurableSpace β
inst✝⁷ : BorelSpace β
inst✝⁶ : TopologicalSpace γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
s : Set α
h : ∀ x ∈ s, s ∈ 𝓝[>] x
x₀ : α
x₀s : x₀ ∈ s
h₀ : IsTop x₀
this : s = {x₀} ∪ s \ {x₀}
x : α
hx : x ∈ s \ {x₀}
⊢ x ≠ x₀

Next tactic:
simpa using hx.2