-- Mathlib/Analysis/Calculus/Monotone.lean
theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
    ∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x

Proof state:
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
⊢ Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)

Tactic:
apply Tendsto.congr' _
  ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))

Result:
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
⊢ (ENNReal.toReal ∘ (fun a => f.measure a / volume a) ∘ fun y => Icc x y) =ᶠ[𝓝[>] x] fun y => (↑f y - ↑f x) / (y - x)

Proof state:
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hxy : y < x
⊢ (ENNReal.toReal ∘ (fun a => f.measure a / volume a) ∘ fun y => Icc y x) y = (leftLim (↑f) y - ↑f x) / (y - x)

Tactic:
simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]

Result:
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hxy : y < x
⊢ (ENNReal.ofReal (↑f x - leftLim (↑f) y) / ENNReal.ofReal (x - y)).toReal = (leftLim (↑f) y - ↑f x) / (y - x)

Proof state:
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
y : ℝ
hxy : x < y
⊢ 0 ≤ (↑f y - ↑f x) / (y - x)

Tactic:
exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le

Result:
no goals

Proof state:
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y => (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hy : y < x
⊢ y < y + 1 * (y - x) ^ 2

Tactic:
have : ↑0 < (x - y) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)

Result:
case h
f : StieltjesFunction
x : ℝ
hx : Tendsto (fun a => f.measure a / volume a) ((vitaliFamily volume 1).filterAt x) (𝓝 (f.measure.rnDeriv volume x))
h'x : f.measure.rnDeriv volume x < ⊤
h''x : ¬leftLim (↑f) x ≠ ↑f x
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
L3 :
  Tendsto (fun y => (leftLim (↑f) (y + 1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[<] x)
    (𝓝 (f.measure.rnDeriv volume x).toReal)
y : ℝ
hy : y < x
this : 0 < (x - y) ^ 2
⊢ y < y + 1 * (y - x) ^ 2