awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
i : (↑s : Type u_2)
⊢ MeasurableSet (u' (↑i : β))

Tactic:
simp_all
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
funProp : ∀ (i : (↑s : Type u_2)), MeasurableSet (u' (↑i : β))
⊢ Measurable f'

Tactic:
fun_prop (disch := simp_all)
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
⊢ Measurable f'

Tactic:
have f'_meas : Measurable f' := by fun_prop (disch := simp_all)
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
        (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)) :
      ℝ≥0∞)

Tactic:
(measure_iUnion_le _).trans ?_

Premises used: MeasureTheory.measure_iUnion_le, LE.le.trans
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
refine :
  LE.le (α := ℝ≥0∞)
    (∑' (i : (↑s : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (⋃ (q : (↑(s ∩ Ioi (↑i : β)) : Type u_2)), u' (↑i : β) ∩ v (↑i : β) (↑q : β)))
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)))
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
        (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)) :
      ℝ≥0∞)

Tactic:
refine (measure_iUnion_le _).trans ?_

Premises used: MeasureTheory.measure_iUnion_le, LE.le.trans
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
⊢ LE.le (α := ℝ≥0∞)
    (∑' (i : (↑s : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (⋃ (q : (↑(s ∩ Ioi (↑i : β)) : Type u_2)), u' (↑i : β) ∩ v (↑i : β) (↑q : β)))
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)))

Tactic:
ENNReal.tsum_le_tsum fun p => ?_

Premises used: ENNReal.tsum_le_tsum
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
refine :
  ∀ (p : (↑s : Type u_2)),
    LE.le (α := ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)) : ℝ≥0∞)
      (∑' (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞)
    (∑' (i : (↑s : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (⋃ (q : (↑(s ∩ Ioi (↑i : β)) : Type u_2)), u' (↑i : β) ∩ v (↑i : β) (↑q : β)))
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)))

Tactic:
refine ENNReal.tsum_le_tsum fun p => ?_

Premises used: ENNReal.tsum_le_tsum
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
p : (↑s : Type u_2)
⊢ Countable.{u_2 + 1} (↑(s ∩ Ioi (↑p : β)) : Type u_2)

Tactic:
haveI := (s_count.mono (s.inter_subset_left (t := Ioi ↑p))).to_subtype
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this_1 : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
p : (↑s : Type u_2)
this : Countable.{u_2 + 1} (↑(s ∩ Ioi (↑p : β)) : Type u_2)
⊢ LE.le (α := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞) (⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)) : ℝ≥0∞)
    (∑' (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)) : ℝ≥0∞)

Tactic:
apply measure_iUnion_le

Premises used: MeasureTheory.measure_iUnion_le
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
h₁ : ∀ (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ⊆ u (↑p : β) (↑q : β)
⊢ LE.le (α := ℝ≥0∞)
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u' (↑p : β) ∩ v (↑p : β) (↑q : β)))
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u (↑p : β) (↑q : β) ∩ v (↑p : β) (↑q : β)))

Tactic:
gcongr with p q
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
p : (↑s : Type u_2)
q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)
⊢ u' (↑p : β) ⊆ u (↑p : β) (↑q : β)

Tactic:
biInter_subset_of_mem q.2

Premises used: Set.biInter_subset_of_mem, Subtype.property
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
p : (↑s : Type u_2)
q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)
⊢ u' (↑p : β) ⊆ u (↑p : β) (↑q : β)

Tactic:
exact biInter_subset_of_mem q.2

Premises used: Set.biInter_subset_of_mem, Subtype.property
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
⊢ Eq (α := ℝ≥0∞)
    (∑' (p : (↑s : Type u_2)) (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)),
      (μ : (a : Set α) → ℝ≥0∞) (u (↑p : β) (↑q : β) ∩ v (↑p : β) (↑q : β)))
    (∑' (p : (↑s : Type u_2)) (x : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), 0)

Tactic:
grind
-- Module: Mathlib.MeasureTheory.Function.AEMeasurableOrder
Goal:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_2
inst : CompleteLinearOrder β
inst_1 : DenselyOrdered β
inst_2 : TopologicalSpace β
inst_3 : OrderTopology β
inst_4 : SecondCountableTopology β
inst_5 : MeasurableSpace β
inst_6 : BorelSpace β
s : Set β
s_count : s.Countable
s_dense : Dense s
f : α → β
h :
  ∀ p ∈ s,
    ∀ q ∈ s,
      p < q →
        ∃ (u : Set α) (v : Set α),
          MeasurableSet u ∧
            MeasurableSet v ∧
              {x : α | f x < p} ⊆ u ∧
                {x : α | q < f x} ⊆ v ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u ∩ v) : ℝ≥0∞) (0 : ℝ≥0∞)
this : Encodable (↑s : Type u_2)
u v : (p q : β) → Set α
huv :
  ∀ (p q : β),
    MeasurableSet (u p q) ∧
      MeasurableSet (v p q) ∧
        {x : α | f x < p} ⊆ u p q ∧
          {x : α | q < f x} ⊆ v p q ∧
            (p ∈ s → q ∈ s → p < q → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (u p q ∩ v p q) : ℝ≥0∞) (0 : ℝ≥0∞))
u' : β → Set α := fun (p : β) ↦ ⋂ q ∈ s ∩ Ioi p, u p q
u'_meas : ∀ (i : β), MeasurableSet (u' i)
f' : α → β :=
  fun (x : α) ↦
    ⨅ (i : (↑s : Type u_2)), (u' (↑i : β)).piecewise (β := fun (x : α) ↦ β) (fun (x : α) ↦ (↑i : β)) (fun (x : α) ↦ ⊤) x
f'_meas : Measurable f'
t : Set α := ⋃ (p : (↑s : Type u_2)), ⋃ (q : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), u' (↑p : β) ∩ v (↑p : β) (↑q : β)
⊢ Eq (α := ℝ≥0∞) (∑' (p : (↑s : Type u_2)) (x : (↑(s ∩ Ioi (↑p : β)) : Type u_2)), 0) 0

Tactic:
simp only [tsum_zero]

Premises used: tsum_zero