-- 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
⊢ ∀ᵐ (x : ℝ), HasDerivAt (↑f) (f.measure.rnDeriv volume x).toReal x

Tactic:
filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (vitaliFamily (volume : Measure ℝ) 1) f.measure,
  rnDeriv_lt_top f.measure volume, f.countable_leftLim_ne.ae_not_mem volume] with x hx h'x h''x

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
⊢ HasDerivAt (↑f) (f.measure.rnDeriv volume x).toReal 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)
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)
⊢ ∀ a ∈ Iio x, (↑f a - ↑f x) / (a - x) ≤ (leftLim (↑f) a - ↑f x) / (a - x)

Tactic:
rintro y (hy : y < x)

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
⊢ (↑f y - ↑f x) / (y - x) ≤ (leftLim (↑f) y - ↑f x) / (y - 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
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)
⊢ Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[<] x) (𝓝 (f.measure.rnDeriv volume x).toReal)

Tactic:
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2

Result:
case hgf
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)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (leftLim (↑f) (b + 1 * (b - x) ^ 2) - ↑f x) / (b - x) ≤ (↑f b - ↑f x) / (b - x)

case hfh
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)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, (↑f b - ↑f x) / (b - x) ≤ (leftLim (↑f) b - ↑f x) / (b - 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
L1 : Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[>] x) (𝓝 (f.measure.rnDeriv volume x).toReal)
⊢ (ENNReal.toReal ∘ (fun a => f.measure a / volume a) ∘ fun y => Icc y x) =ᶠ[𝓝[<] x] fun y =>
    (leftLim (↑f) y - ↑f x) / (y - x)

Tactic:
filter_upwards [self_mem_nhdsWithin]

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)
⊢ ∀ a ∈ Iio x,
    (ENNReal.toReal ∘ (fun a => f.measure a / volume a) ∘ fun y => Icc y x) a = (leftLim (↑f) a - ↑f x) / (a - 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
⊢ HasDerivAt (↑f) (f.measure.rnDeriv volume x).toReal x

Tactic:
have L1 :
  Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (rnDeriv f.measure volume x).toReal) := by
  apply Tendsto.congr' _
    ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))
  filter_upwards [self_mem_nhdsWithin]
  rintro y (hxy : x < y)
  simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]
  rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]
  exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le

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)
⊢ HasDerivAt (↑f) (f.measure.rnDeriv volume x).toReal 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)
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
⊢ ↑f y ≤ leftLim (↑f) (y + 1 * (y - x) ^ 2)

Tactic:
apply f.mono.le_leftLim

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
⊢ y < y + 1 * (y - x) ^ 2