smooth

formal source

  90
  91noncomputable section
  92
  93private abbrev smooth : WithTop ℕ∞ := (⊤ : ℕ∞)
  94
  95private def Phi (H : ℝ → ℝ) (t : ℝ) : ℝ := ∫ s in (0 : ℝ)..t, H s
  96
  97private lemma phi_zero (H : ℝ → ℝ) : Phi H 0 = 0 := by
  98  simp [Phi, intervalIntegral.integral_same]
  99
 100private lemma phi_hasDerivAt (H : ℝ → ℝ) (h_cont : Continuous H) (t : ℝ) :
 101    HasDerivAt (Phi H) (H t) t :=
 102  intervalIntegral.integral_hasDerivAt_right (h_cont.intervalIntegrable 0 t)
 103    h_cont.aestronglyMeasurable.stronglyMeasurableAtFilter h_cont.continuousAt
 104
 105private lemma phi_differentiable (H : ℝ → ℝ) (h_cont : Continuous H) :
 106    Differentiable ℝ (Phi H) :=
 107  fun t => (phi_hasDerivAt H h_cont t).differentiableAt
 108
 109private lemma deriv_phi_eq (H : ℝ → ℝ) (h_cont : Continuous H) : deriv (Phi H) = H :=
 110  funext fun t => (phi_hasDerivAt H h_cont t).deriv
 111
 112private lemma exists_integral_ne_zero (H : ℝ → ℝ) (h_one : H 0 = 1) (h_cont : Continuous H) :
 113    ∃ δ : ℝ, 0 < δ ∧ Phi H δ ≠ 0 := by
 114  have h_pos : (0 : ℝ) < H 0 := by rw [h_one]; exact one_pos
 115  have h_ev : ∀ᶠ x in nhds (0 : ℝ), (0 : ℝ) < H x :=
 116    h_cont.continuousAt.eventually (Ioi_mem_nhds h_pos)
 117  obtain ⟨ε, hε_pos, hε⟩ := Metric.eventually_nhds_iff.mp h_ev
 118  refine ⟨ε / 2, by positivity, ?_⟩
 119  intro h_eq
 120  have hδ_pos : (0 : ℝ) < ε / 2 := by positivity
 121  obtain ⟨c, hc_mem, hc_eq⟩ := exists_hasDerivAt_eq_slope (Phi H) H hδ_pos
 122    ((phi_differentiable H h_cont).continuous.continuousOn)
 123    (fun x _ => phi_hasDerivAt H h_cont x)