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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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; linarith
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
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