awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : y ∈ Ioi x
⊢ 0 < y - x

Tactic:
rw [mem_Ioi, ← sub_pos] at hy

Premises used: sub_pos, Set.mem_Ioi
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
h : (↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) ≤ f y
⊢ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x) ≤ (f y - f x) / (y - x)

Tactic:
gcongr
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
nlinarith :
  y ≤ y + -1 * HPow.hPow (β := ℕ) (y - x) 2 →
    y - (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) + (0 - (0 - (y - x)) * (0 - (y - x))) = 0
⊢ y + -1 * HPow.hPow (β := ℕ) (y - x) 2 < y

Tactic:
nlinarith
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
⊢ (↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) ≤ f y

Tactic:
exact hf.rightLim_le (by nlinarith)

Premises used: Monotone.rightLim_le
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h : ∀ (y : ℝ) (hy : y ∈ Ioi x), (f y - f x) / (y - x) ≤ ((↑hf.stieltjesFunction : ℝ → ℝ) y - f x) / (y - x)
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] x, (f b - f x) / (b - x) ≤ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)

Tactic:
filter_upwards [self_mem_nhdsWithin] with y hy

Premises used: self_mem_nhdsWithin
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
h : f y ≤ (↑hf.stieltjesFunction : ℝ → ℝ) y
⊢ (f y - f x) / (y - x) ≤ ((↑hf.stieltjesFunction : ℝ → ℝ) y - f x) / (y - x)

Tactic:
gcongr
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
⊢ f y ≤ (↑hf.stieltjesFunction : ℝ → ℝ) y

Tactic:
hf.le_rightLim le_rfl

Premises used: le_rfl, Function.rightLim, Monotone.le_rightLim, Function.leftLim, ite
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[>] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : 0 < y - x
⊢ f y ≤ (↑hf.stieltjesFunction : ℝ → ℝ) y

Tactic:
exact hf.le_rightLim le_rfl

Premises used: le_rfl, Monotone.le_rightLim
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
⊢ Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)

Tactic:
have L1 : Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x)
      (𝓝 (rnDeriv hf.stieltjesFunction.measure volume x).toReal) := by
    -- limit of a helper function, with a small shift compared to `g`
    have : Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[>] x)
        (𝓝 (rnDeriv hf.stieltjesFunction.measure volume x).toReal) := by
      apply tendsto_apply_add_mul_sq_div_sub (nhdsGT_le_nhdsNE x) hx.2
      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
      · apply Tendsto.mono_left _ nhdsWithin_le_nhds
        have : Tendsto (fun y : ℝ => y + -↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -↑1 * (x - x) ^ 2)) :=
          tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))
        simpa using this
      · filter_upwards [Ioo_mem_nhdsGT <| show x < x + 1 by simp]
        rintro y ⟨hy : x < y, h'y : y < x + 1⟩
        rw [mem_Ioi]
        nlinarith
    -- apply the sandwiching argument, with the helper function and `g`
    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' this hx.2
    · filter_upwards [self_mem_nhdsWithin] with y hy
      rw [mem_Ioi, ← sub_pos] at hy
      gcongr
      exact hf.rightLim_le (by nlinarith)
    · filter_upwards [self_mem_nhdsWithin] with y hy
      rw [mem_Ioi, ← sub_pos] at hy
      gcongr
      exact hf.le_rightLim le_rfl
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
apply : Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝[<] x) (𝓝[<] x)
⊢ Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[<] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)

Tactic:
apply tendsto_apply_add_mul_sq_div_sub (nhdsLT_le_nhdsNE x) hx.1

Premises used: And.left, nhdsLT_le_nhdsNE, tendsto_apply_add_mul_sq_div_sub
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h1 : Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝[<] x) (𝓝 x)
h2 : ∀ᶠ (x_1 : ℝ) in 𝓝[<] x, x_1 + -1 * HPow.hPow (β := ℕ) (x_1 - x) 2 ∈ Iio x
⊢ Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝[<] x) (𝓝[<] x)

Tactic:
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within

Premises used: tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
apply : Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝 x) (𝓝 x)
⊢ Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝[<] x) (𝓝 x)

Tactic:
apply Tendsto.mono_left _ nhdsWithin_le_nhds

Premises used: nhdsWithin_le_nhds, Filter.Tendsto.mono_left
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
⊢ Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝 x) (𝓝 (x + -1 * HPow.hPow (β := ℕ) (x - x) 2))

Tactic:
have : Tendsto (fun y : ℝ => y + -↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -↑1 * (x - x) ^ 2)) :=
          tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
this : Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝 x) (𝓝 (x + -1 * HPow.hPow (β := ℕ) (x - x) 2))
⊢ Tendsto (fun (y : ℝ) ↦ y + -1 * HPow.hPow (β := ℕ) (y - x) 2) (𝓝 x) (𝓝 x)

Tactic:
simpa using this
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
⊢ x - 1 < x

Tactic:
simp
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h : ∀ a ∈ Ioo (x - 1) x, a + -1 * HPow.hPow (β := ℕ) (a - x) 2 ∈ Iio x
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[<] x, x_1 + -1 * HPow.hPow (β := ℕ) (x_1 - x) 2 ∈ Iio x

Tactic:
filter_upwards [Ioo_mem_nhdsLT <| show x - 1 < x by simp]

Premises used: Ioo_mem_nhdsLT
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : y ∈ Ioo (x - 1) x
⊢ x - 1 < y ∧ y < x

Tactic:
rw [mem_Ioo] at hy

Premises used: Set.mem_Ioo
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
y : ℝ
hy : x - 1 < y ∧ y < x
nlinarith :
  x ≤ y + -1 * HPow.hPow (β := ℕ) (y - x) 2 →
    0 - HPow.hPow (β := ℕ) (y - x) 2 + (y - x) + (x - (y + -1 * HPow.hPow (β := ℕ) (y - x) 2)) = 0
⊢ y + -1 * HPow.hPow (β := ℕ) (y - x) 2 < x

Tactic:
nlinarith
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
⊢ Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[<] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)

Tactic:
have : Tendsto (fun y => (hf.stieltjesFunction (y + -1 * (y - x) ^ 2) - f x) / (y - x)) (𝓝[<] x)
        (𝓝 (rnDeriv hf.stieltjesFunction.measure volume x).toReal) := by
      apply tendsto_apply_add_mul_sq_div_sub (nhdsLT_le_nhdsNE x) hx.1
      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
      · apply Tendsto.mono_left _ nhdsWithin_le_nhds
        have : Tendsto (fun y : ℝ => y + -↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + -↑1 * (x - x) ^ 2)) :=
          tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul (-1))
        simpa using this
      · filter_upwards [Ioo_mem_nhdsLT <| show x - 1 < x by simp]
        rintro y hy
        rw [mem_Ioo] at hy
        rw [mem_Iio]
        nlinarith
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[<] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
hgf : ∀ᶠ (b : ℝ) in 𝓝[<] x, ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x) ≤ (f b - f x) / (b - x)
hfh :
  ∀ᶠ (b : ℝ) in 𝓝[<] x,
    (f b - f x) / (b - x) ≤ ((↑hf.stieltjesFunction : ℝ → ℝ) (b + -1 * HPow.hPow (β := ℕ) (b - x) 2) - f x) / (b - x)
⊢ Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[<] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)

Tactic:
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hx.1 this

Premises used: tendsto_of_tendsto_of_tendsto_of_le_of_le', And.left
-- Module: Mathlib.Analysis.Calculus.Monotone
Goal:
f : ℝ → ℝ
hf : Monotone f
x : ℝ
hx :
  Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[<] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal) ∧
    Tendsto (fun (b : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x)) (𝓝[>] x)
      (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h'x : ¬¬ContinuousAt f x
A : (↑hf.stieltjesFunction : ℝ → ℝ) x = f x
L1 :
  Tendsto (fun (y : ℝ) ↦ (f y - f x) / (y - x)) (𝓝[>] x)
    (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
this :
  Tendsto (fun (y : ℝ) ↦ ((↑hf.stieltjesFunction : ℝ → ℝ) (y + -1 * HPow.hPow (β := ℕ) (y - x) 2) - f x) / (y - x))
    (𝓝[<] x) (𝓝 (rnDeriv (m := Real.measurableSpace) (StieltjesFunction.measure hf.stieltjesFunction) volume x).toReal)
h : ∀ a ∈ Iio x, ((↑hf.stieltjesFunction : ℝ → ℝ) a - f x) / (a - x) ≤ (f a - f x) / (a - x)
⊢ ∀ᶠ (b : ℝ) in 𝓝[<] x, ((↑hf.stieltjesFunction : ℝ → ℝ) b - f x) / (b - x) ≤ (f b - f x) / (b - x)

Tactic:
filter_upwards [self_mem_nhdsWithin]

Premises used: self_mem_nhdsWithin