| import OAI.Probability.FirstPassage.GlobalDifferentiability |
| import Mathlib.Analysis.Calculus.Darboux |
| import Mathlib.Topology.Order.MonotoneContinuity |
| import Mathlib.Analysis.Calculus.ContDiff.Deriv |
| import Mathlib.Analysis.Calculus.ContDiff.Operations |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Set Filter |
| open scoped Topology |
|
|
| lemma convex_continuous_deriv (φ : ℝ → ℝ) (hc : ConvexOn ℝ univ φ) |
| (hd : Differentiable ℝ φ) : Continuous (deriv φ) := by |
| let g := fun x => deriv φ x+x |
| have hm : Monotone g := fun x y h => add_le_add |
| (hc.monotoneOn_deriv (fun x _ => hd x) (mem_univ x) (mem_univ y) h) h |
| have hg (x : ℝ) : HasDerivAt (fun t => φ t+t^2/2) (g x) x := by |
| have hq : HasDerivAt (fun t : ℝ => t^2/2) x x := by |
| simpa using ((hasDerivAt_id x).pow 2).div_const 2 |
| exact (hd x).hasDerivAt.add hq |
| have hsurj : Function.Surjective g := by |
| intro y |
| let a := min 0 (y-deriv φ 0-1) |
| let b := max 0 (y-deriv φ 0+1) |
| have ha : a ≤ 0 := min_le_left .. |
| have hb : 0 ≤ b := le_max_left .. |
| have hda := hc.monotoneOn_deriv (fun x _ => hd x) (mem_univ a) (mem_univ 0) ha |
| have hdb := hc.monotoneOn_deriv (fun x _ => hd x) (mem_univ 0) (mem_univ b) hb |
| have hay : a ≤ y-deriv φ 0-1 := min_le_right .. |
| have hby : y-deriv φ 0+1 ≤ b := le_max_right .. |
| have hga : g a < y := by dsimp [g]; linarith |
| have hgb : y < g b := by dsimp [g]; linarith |
| obtain ⟨x,_hx,hx⟩ := exists_hasDerivWithinAt_eq_of_gt_of_lt (ha.trans hb) |
| (fun x _ => (hg x).hasDerivWithinAt) hga hgb |
| exact ⟨x,hx⟩ |
| have hcont := hm.continuous_of_surjective hsurj |
| convert hcont.sub continuous_id using 1 |
| funext x |
| dsimp [g] |
| ring |
|
|
| lemma convex_contDiff_one (φ : ℝ → ℝ) (hc : ConvexOn ℝ univ φ) |
| (hd : Differentiable ℝ φ) : ContDiff ℝ 1 φ := |
| contDiff_one_iff_deriv.mpr ⟨hd,convex_continuous_deriv φ hc hd⟩ |
|
|
| end PlanarFPP |
|
|
| end OAI |
|
|
