Phi

formal source

  28
  29/-! ## Phase 1: Integration Bootstrap (Continuous → C^∞) -/
  30
  31private def Phi (H : ℝ → ℝ) (t : ℝ) : ℝ := ∫ s in (0 : ℝ)..t, H s
  32
  33private lemma phi_zero (H : ℝ → ℝ) : Phi H 0 = 0 := by
  34  simp [Phi, intervalIntegral.integral_same]
  35
  36private lemma phi_hasDerivAt (H : ℝ → ℝ) (h_cont : Continuous H) (t : ℝ) :
  37    HasDerivAt (Phi H) (H t) t :=
  38  intervalIntegral.integral_hasDerivAt_right (h_cont.intervalIntegrable 0 t)
  39    h_cont.aestronglyMeasurable.stronglyMeasurableAtFilter h_cont.continuousAt
  40
  41private lemma phi_differentiable (H : ℝ → ℝ) (h_cont : Continuous H) :
  42    Differentiable ℝ (Phi H) :=
  43  fun t => (phi_hasDerivAt H h_cont t).differentiableAt
  44
  45private lemma deriv_phi_eq (H : ℝ → ℝ) (h_cont : Continuous H) : deriv (Phi H) = H :=
  46  funext fun t => (phi_hasDerivAt H h_cont t).deriv
  47
  48private lemma exists_integral_ne_zero (H : ℝ → ℝ) (h_one : H 0 = 1) (h_cont : Continuous H) :
  49    ∃ δ : ℝ, 0 < δ ∧ Phi H δ ≠ 0 := by
  50  have h_pos : (0 : ℝ) < H 0 := by rw [h_one]; exact one_pos
  51  have h_ev : ∀ᶠ x in nhds (0 : ℝ), (0 : ℝ) < H x :=
  52    h_cont.continuousAt.eventually (Ioi_mem_nhds h_pos)
  53  obtain ⟨ε, hε_pos, hε⟩ := Metric.eventually_nhds_iff.mp h_ev
  54  refine ⟨ε / 2, by positivity, ?_⟩
  55  intro h_eq
  56  have hδ_pos : (0 : ℝ) < ε / 2 := by positivity
  57  obtain ⟨c, hc_mem, hc_eq⟩ := exists_hasDerivAt_eq_slope (Phi H) H hδ_pos
  58    ((phi_differentiable H h_cont).continuous.continuousOn)
  59    (fun x _ => phi_hasDerivAt H h_cont x)
  60  rw [phi_zero, h_eq, sub_zero, zero_div] at hc_eq
  61  linarith [hε (show dist c 0 < ε by