-- Mathlib/Analysis/Calculus/Monotone.lean
theorem Monotone.ae_hasDerivAt {f : ℝ → ℝ} (hf : Monotone f) :
    ∀ᵐ x, HasDerivAt f (rnDeriv hf.stieltjesFunction.measure volume x).toReal x

Proof state:
case h
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x

Tactic:
rw [hasDerivAt_iff_tendsto_slope, slope_fun_def_field, ← nhdsLT_sup_nhdsGT, tendsto_sup]

Result:
case h
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
L2 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun b => (f b - f x) / (b - x)) (𝓝[<] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (f b - f x) / (b - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Proof state:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : ¬¬ContinuousAt f x
⊢ ↑hf.stieltjesFunction x = f x

Tactic:
rw [Classical.not_not, hf.continuousAt_iff_leftLim_eq_rightLim] at h'x

Result:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : leftLim f x = rightLim f x
⊢ ↑hf.stieltjesFunction x = f x

Proof state:
case h.hab.h
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
this :
  Tendsto (fun y => (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
y : ℝ
hy : 0 < y - x
⊢ ↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) ≤ f y

Tactic:
exact hf.rightLim_le (by nlinarith)

Result:
no goals

Proof state:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
this✝ :
  Tendsto (fun y => (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
y : ℝ
hy : y - x < 0
this : 0 < (y - x) ^ 2
⊢ y + -1 * (y - x) ^ 2 < y

Tactic:
linarith

Result:
no goals

Proof state:
case h
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx : HasDerivAt (↑hf.stieltjesFunction) (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
⊢ HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x

Tactic:
rw [hasDerivAt_iff_tendsto_slope, (nhdsLT_sup_nhdsGT x).symm, tendsto_sup,
  slope_fun_def_field, A] at hx

Result:
case h
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
⊢ HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv volume x).toReal x

Proof state:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun y => (↑hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)

Tactic:
apply tendsto_apply_add_mul_sq_div_sub (nhdsLT_le_nhdsNE x) hx.1

Result:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ Tendsto (fun y => y + -1 * (y - x) ^ 2) (𝓝[<] x) (𝓝[<] x)

Proof state:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧
    Tendsto (fun b => (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : ↑hf.stieltjesFunction x = f x
L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal)
⊢ x - 1 < x

Tactic:
simp

Result:
no goals