| import OAI.Geometry.IsometricImmersion.Caps.IntegratedCapOverlap |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory |
| open scoped ContDiff Topology Interval |
|
|
| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry SmoothLocal.Flow |
|
|
| def capInteriorWeightFloor (top b lambda MI : ℝ) : ℝ := |
| (b-top)^8*Real.exp (-lambda*top-MI) |
|
|
| theorem capInteriorWeightFloor_pos {top b lambda MI : ℝ} (htop : top < b) : |
| 0 < capInteriorWeightFloor top b lambda MI := |
| mul_pos (pow_pos (sub_pos.mpr htop) _) (Real.exp_pos _) |
|
|
| theorem directedWeight_interior_lower {I : Coord → ℝ} {p : Coord} {top b lambda MI : ℝ} |
| (htop : top < b) (hlambda : 0 ≤ lambda) (hp : p 1 ≤ top) (hI : |I p| ≤ MI) : |
| capInteriorWeightFloor top b lambda MI ≤ directedWeight b lambda I p := by |
| have hdist : b-top ≤ edgeDistance b p := by unfold edgeDistance |
| have hpow := pow_le_pow_left₀ (sub_pos.mpr htop).le hdist 8 |
| have hphase : -lambda*top-MI ≤ weightPhase lambda I p := by |
| have hm := mul_le_mul_of_nonneg_left hp hlambda |
| unfold weightPhase |
| linarith [(abs_le.mp hI).1] |
| exact mul_le_mul hpow (Real.exp_le_exp.mpr hphase) (Real.exp_pos _).le |
| (pow_nonneg ((sub_pos.mpr htop).le.trans hdist) _) |
|
|
| theorem closedRectangle_measurable (tl tr sb st : ℝ) : MeasurableSet (closedRectangle tl tr sb st) := by |
| have h0 : IsClosed ((fun p : Coord => p 0) ⁻¹' Icc tl tr) := isClosed_Icc.preimage (continuous_apply 0) |
| have h1 : IsClosed ((fun p : Coord => p 1) ⁻¹' Icc sb st) := isClosed_Icc.preimage (continuous_apply 1) |
| exact (h0.inter h1).measurableSet |
|
|
| theorem coordinateSubsetIntegral_closedRectangle (tl tr sb st : ℝ) (F : Coord → ℝ) |
| (ht : tl ≤ tr) (hs : sb ≤ st) (hF : ContinuousOn F (closedRectangle tl tr sb st)) : |
| coordinateSubsetIntegral (closedRectangle tl tr sb st) F = rectangleIntegral tl tr sb st F := by |
| rw [rectangleIntegral_eq_closed_area ht hs hF] |
| have hpre : (fun q : ℝ × ℝ => boxPoint q.1 q.2) ⁻¹' closedRectangle tl tr sb st = |
| Icc tl tr ×ˢ Icc sb st := by |
| ext q |
| simp [closedRectangle,boxPoint,Prod.le_def,and_left_comm,and_assoc] |
| unfold coordinateSubsetIntegral |
| rw [hpre] |
| |
| theorem capInterior_gradient_le_directed_energy |
| {L R bottom top b lambda MI : ℝ} {I u : Coord → ℝ} |
| (hL : -2 < L) (hR : R < 2) (hbottom : -2 < bottom) |
| (hLR : L ≤ R) (hbt : bottom ≤ top) (htop : top < b) (hb : b ≤ 0) |
| (hlambda : 0 ≤ lambda) (hI : ContDiffOn ℝ ∞ I capChartDomain) |
| (hu : ContDiffOn ℝ ∞ u capChartDomain) |
| (hIv : ∀ p ∈ closedRectangle L R bottom top, |I p| ≤ MI) : |
| rectangleIntegral L R bottom top (fun p => (coordPartial 0 u p)^2+(coordPartial 1 u p)^2) ≤ |
| (1/capInteriorWeightFloor top b lambda MI)* |
| rectangleIntegral (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b |
| (directedGradientEnergy (capSpatialCutoff L R bottom) I u b lambda) := by |
| let F := fun p => (coordPartial 0 u p)^2+(coordPartial 1 u p)^2 |
| let E := directedGradientEnergy (capSpatialCutoff L R bottom) I u b lambda |
| let S := closedRectangle L R bottom top |
| let T := closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom bottom) b |
| have hwpos := capInteriorWeightFloor_pos (lambda := lambda) (MI := MI) htop |
| obtain ⟨htlo,htwidth,htone,hrone,hrwidth,htrhi,hsblo,hswidth,hsone,hlt,hrt⟩ := |
| capOuterRectangle_edges hL hR hbottom |
| have ht : capOuterLeft L ≤ capOuterRight R := by linarith |
| have hs : capOuterBottom bottom ≤ b := by linarith |
| have hT : T ⊆ capChartDomain := capOuterRectangle_subset_domain hL hR hbottom (by linarith) |
| have hST : S ⊆ T := by |
| intro p hp |
| exact ⟨⟨hlt.le.trans hp.1.1,hp.1.2.trans hrt.le⟩, |
| ⟨(by linarith [hp.2.1]),hp.2.2.trans htop.le⟩⟩ |
| have hgrad := ((partial_contDiffOn hu capChartDomain_isOpen 0).pow 2).add |
| ((partial_contDiffOn hu capChartDomain_isOpen 1).pow 2) |
| have hFc : ContinuousOn F T := hgrad.continuousOn.mono hT |
| have hEc : ContinuousOn E T := |
| ((((capSpatialCutoff_contDiff L R bottom).contDiffOn.mul |
| (directedWeight_contDiffOn b lambda hI)).mul hgrad).continuousOn.mono hT) |
| have hEn (p : Coord) (hp : p ∈ T) : 0 ≤ E p := by |
| have hchi := (capSpatialCutoff_range L R bottom p).1 |
| have hd : 0 ≤ edgeDistance b p := sub_nonneg.mpr hp.2.2 |
| unfold E directedGradientEnergy directedWeight |
| positivity |
| have hpoint (p : Coord) (hp : p ∈ S) : F p ≤ (1/capInteriorWeightFloor top b lambda MI)*E p := by |
| have hW := directedWeight_interior_lower htop hlambda hp.2.2 (hIv p hp) |
| have hchi := capSpatialCutoff_one hL hR hbottom p ⟨hp.1.1,hp.1.2,hp.2.1⟩ |
| have hgrad0 : 0 ≤ F p := add_nonneg (sq_nonneg _) (sq_nonneg _) |
| have hh := mul_le_mul_of_nonneg_right hW hgrad0 |
| have hscaled : capInteriorWeightFloor top b lambda MI*F p ≤ E p := by |
| simpa only [E,directedGradientEnergy,hchi,one_mul] using hh |
| have hdiv : F p ≤ E p/capInteriorWeightFloor top b lambda MI := |
| (le_div_iff₀ hwpos).mpr (by simpa only [mul_comm] using hscaled) |
| convert hdiv using 1 |
| ring |
| have hcomp := restricted_gradient_from_energy ht hs (closedRectangle_measurable _ _ _ _) |
| hST hFc hEc hEn (show 0 ≤ 1/capInteriorWeightFloor top b lambda MI by positivity) hpoint |
| rw [coordinateSubsetIntegral_closedRectangle L R bottom top F hLR hbt (hFc.mono hST)] at hcomp |
| exact hcomp |
| |
| end SmoothLocal.Weighted |
| |
| end |
| |
| end OAI |
| |
