awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
inl :
  ∀ (hR : IsEmpty R),
    ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have inr : ∀ (hR : Nonempty R),
  ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
Tactic: rcases isEmpty_or_nonempty R with hR | hR
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : IsEmpty R
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have inl : t i ⊆ ∅
Tactic: refine ⟨∅, ?_, MeasurableSet.empty, by simp⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have hl : f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
Tactic: have hl :=
        ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl : f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
| f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
have lhs : f.length (t i)
Tactic: lhs
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl : f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
| f.length (t i)
have rw : ⨅ (a : R), ⨅ (b : R), ⨅ (_ : t i \ botSet ⊆ Ioc a b), ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a)
Tactic: rw [length_eq]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl : f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
| f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
have rw : ⨅ (a : R), ⨅ (b : R), ⨅ (_ : t i \ botSet ⊆ Ioc a b), ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a)
Tactic: lhs
        rw [length_eq]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl : f.length (t i) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have hl₁ : ⨅ (a : R), ⨅ (b : R), ⨅ (_ : t i \ botSet ⊆ Ioc a b), ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a) <
  f.length (t i) + (↑(ε' i) : ℝ≥0∞)
Tactic: conv at hl =>
        lhs
        rw [length_eq]
Cleared hypotheses: hl
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl :
  ⨅ (a : R), ⨅ (b : R), ⨅ (_ : t i \ botSet ⊆ Ioc a b), ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a) <
    f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have hl₁ : ∃ (i_1 : R) (i_2 : R) (_ : t i \ botSet ⊆ Ioc i_1 i_2),
  ofReal ((↑f : R → ℝ) i_2 - (↑f : R → ℝ) i_1) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
Tactic: simp only [iInf_lt_iff] at hl
Cleared hypotheses: hl
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
hl :
  ∃ (i_1 : R) (i_2 : R) (_ : t i \ botSet ⊆ Ioc i_1 i_2),
    ofReal ((↑f : R → ℝ) i_2 - (↑f : R → ℝ) i_1) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have inr : ∀ (a b : R) (h₁ : t i \ botSet ⊆ Ioc a b)
  (h₂ : ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)),
  ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
Tactic: rcases hl with ⟨a, b, h₁, h₂⟩
Cleared hypotheses: hl
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have h₃ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
Tactic: rw [← f.outer_Ioc] at h₂
Cleared hypotheses: h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i \ botSet ⊆ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have h₃ : t i ⊆ botSet ∪ Ioc a b
Tactic: rw [diff_subset_iff] at h₁
Cleared hypotheses: h₁, h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ ∃ (s : Set R), t i ⊆ s ∧ MeasurableSet s ∧ (f.outer : (a : Set R) → ℝ≥0∞) s ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
have inr : (f.outer : (a : Set R) → ℝ≥0∞) (botSet ∪ Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
Tactic: refine ⟨_, h₁, measurableSet_botSet.union measurableSet_Ioc, le_of_lt ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) botSet : ℝ≥0∞)
      ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) ≤
    HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
have bc : (f.outer : (a : Set R) → ℝ≥0∞) botSet ≤ f.length botSet
Tactic: gcongr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
have simp : f.length (∅ \ botSet) + ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a) = ofReal ((↑f : R → ℝ) b - (↑f : R → ℝ) a)
Tactic: simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ (f.outer : (a : Set R) → ℝ≥0∞) (botSet ∪ Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
have calc : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) botSet : ℝ≥0∞)
    ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) ≤
  HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
Tactic: calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
        simp [empty_diff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
calc :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) botSet : ℝ≥0∞)
      ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) ≤
    HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
⊢ (f.outer : (a : Set R) → ℝ≥0∞) (botSet ∪ Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
have calc₁ : HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
Tactic: calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
        simp [empty_diff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
calc :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) botSet : ℝ≥0∞)
      ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) ≤
    HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
calc₁ :
  HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
⊢ (f.outer : (a : Set R) → ℝ≥0∞) (botSet ∪ Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
have calc₂ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
  (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b)
Tactic: calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
        simp [empty_diff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
i : ℕ
hR : Nonempty R
a b : R
h₁ : t i ⊆ botSet ∪ Ioc a b
h₂ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + (↑(ε' i) : ℝ≥0∞)
calc :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) botSet : ℝ≥0∞)
      ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) ≤
    HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
calc₁ :
  HAdd.hAdd (β := ℝ≥0∞) (f.length botSet) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
    HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞)
calc₂ :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞) ((f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) : ℝ≥0∞) =
    (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b)
⊢ (f.outer : (a : Set R) → ℝ≥0∞) (botSet ∪ Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
have calc₃ : (f.outer : (a : Set R) → ℝ≥0∞) (Ioc a b) < f.length (t i) + ofReal (↑(ε' i) : ℝ)
Tactic: calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
        simp [empty_diff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (⨅ (t : Set R), ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), (f.outer : (a : Set R) → ℝ≥0∞) t)
    (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
have choose : ∀ (g : (i : ℕ) → Set R)
  (hg :
    ∀ (i : ℕ),
      t i ⊆ g i ∧ MeasurableSet (g i) ∧ (f.outer : (a : Set R) → ℝ≥0∞) (g i) ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)),
  LE.le (α := ℝ≥0∞) (⨅ (t : Set R), ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), (f.outer : (a : Set R) → ℝ≥0∞) t)
    (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
Tactic: choose g hg using
    show ∀ i, ∃ s, t i ⊆ s ∧ MeasurableSet s ∧ f.outer s ≤ f.length (t i) + ofReal (ε' i) by
      intro i
      rcases isEmpty_or_nonempty R with hR | hR
      · refine ⟨∅, ?_, MeasurableSet.empty, by simp⟩
        simpa using eq_empty_of_isEmpty (t i)
      have hl :=
        ENNReal.lt_add_right ((ENNReal.le_tsum i).trans_lt h).ne (ENNReal.coe_pos.2 (ε'0 i)).ne'
      conv at hl =>
        lhs
        rw [length_eq]
      simp only [iInf_lt_iff] at hl
      rcases hl with ⟨a, b, h₁, h₂⟩
      rw [← f.outer_Ioc] at h₂
      rw [diff_subset_iff] at h₁
      refine ⟨_, h₁, measurableSet_botSet.union measurableSet_Ioc, le_of_lt ?_⟩
      calc f.outer (botSet ∪ Ioc a b)
      _ ≤ f.outer botSet + f.outer (Ioc a b) := measure_union_le _ _
      _ ≤ f.length botSet + f.outer (Ioc a b) := by gcongr; apply outer_le_length
      _ = 0 + f.outer (Ioc a b) := by
        simp only [← length_diff_botSet, sdiff_self, bot_eq_empty, empty_diff, outer_Ioc, zero_add]
        simp [empty_diff]
      _ = f.outer (Ioc a b) := by simp
      _ < f.length (t i) + ofReal ↑(ε' i) := by simpa using h₂
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (hypothesis)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
g : (i : ℕ) → Set R
hg :
  ∀ (i : ℕ),
    t i ⊆ g i ∧ MeasurableSet (g i) ∧ (f.outer : (a : Set R) → ℝ≥0∞) (g i) ≤ f.length (t i) + ofReal (↑(ε' i) : ℝ)
⊢ LE.le (α := ℝ≥0∞) (⨅ (t : Set R), ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), (f.outer : (a : Set R) → ℝ≥0∞) t)
    (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
have hg₁ : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ (f.outer : (a : Set R) → ℝ≥0∞) (g i) ≤ f.length (t i) + (↑(ε' i) : ℝ≥0∞)
Tactic: simp only [ofReal_coe_nnreal] at hg
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
g : (i : ℕ) → Set R
hg :
  ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ (f.outer : (a : Set R) → ℝ≥0∞) (g i) ≤ f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (⨅ (t : Set R), ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), (f.outer : (a : Set R) → ℝ≥0∞) t)
    (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
have apply : LE.le (α := ℝ≥0∞) (⨅ (_ : s ⊆ iUnion g), ⨅ (_ : MeasurableSet (iUnion g)), (f.outer : (a : Set R) → ℝ≥0∞) (iUnion g))
  (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
Tactic: apply iInf_le_of_le (iUnion g) _
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Stieltjes (goal)
Goal:
R : Type u_1
inst : LinearOrder R
inst_1 : TopologicalSpace R
f : StieltjesFunction R
inst_2 : OrderTopology R
inst_3 : CompactIccSpace R
inst_4 : MeasurableSpace R
inst_5 : BorelSpace R
inst_6 : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ (i : ℕ), t i
ε : ℝ≥0
ε0 : 0 < ε
h : LT.lt (α := ℝ≥0∞) (∑' (i : ℕ), f.length (t i)) ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), (↑(ε' i) : ℝ≥0∞) < (↑ε : ℝ≥0∞)
g : (i : ℕ) → Set R
hg :
  ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ (f.outer : (a : Set R) → ℝ≥0∞) (g i) ≤ f.length (t i) + (↑(ε' i) : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) (⨅ (_ : s ⊆ iUnion g), ⨅ (_ : MeasurableSet (iUnion g)), (f.outer : (a : Set R) → ℝ≥0∞) (iUnion g))
    (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
have apply : LE.le (α := ℝ≥0∞) (⨅ (_ : MeasurableSet (iUnion g)), (f.outer : (a : Set R) → ℝ≥0∞) (iUnion g))
  (∑' (a : ℕ), (f.length (t a) + (↑(ε' a) : ℝ≥0∞)))
Tactic: apply iInf_le_of_le (ht.trans <| iUnion_mono fun i => (hg i).1) _