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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 | import OAI.Geometry.IsometricImmersion.Caps.CapOverlapCosts
namespace OAI
noncomputable section
open Set
open scoped ContDiff
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE
def capOuterLeft (L : ℝ) : ℝ := capOuterEdge (capOuterEdge L)
def capOuterLeftOne (L : ℝ) : ℝ := capInnerEdge (capOuterEdge L)
def capOuterRightOne (R : ℝ) : ℝ := -capInnerEdge (capOuterEdge (-R))
def capOuterRight (R : ℝ) : ℝ := -capOuterEdge (capOuterEdge (-R))
def capOuterBottom (A : ℝ) : ℝ := capOuterEdge (capOuterEdge A)
def capOuterBottomOne (A : ℝ) : ℝ := capInnerEdge (capOuterEdge A)
theorem capOuterSpatialCutoff_eq_threeEdge (L R A : ℝ) :
capOuterSpatialCutoff L R A =
threeEdgeCutoff (capOuterLeft L) (capOuterLeftOne L) (capOuterRightOne R)
(capOuterRight R) (capOuterBottom A) (capOuterBottomOne A) := by
simp only [capOuterSpatialCutoff, capSpatialCutoff, neg_neg, capOuterLeft, capOuterLeftOne,
capOuterRightOne, capOuterRight, capOuterBottom, capOuterBottomOne]
theorem capOuterRectangle_edges {L R A : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) :
-2 < capOuterLeft L ∧ capOuterLeft L < capOuterLeftOne L ∧ capOuterLeftOne L < -3/2 ∧
3/2 < capOuterRightOne R ∧ capOuterRightOne R < capOuterRight R ∧ capOuterRight R < 2 ∧
-2 < capOuterBottom A ∧ capOuterBottom A < capOuterBottomOne A ∧
capOuterBottomOne A < A ∧ capOuterLeft L < L ∧ R < capOuterRight R := by
obtain ⟨hlo,hli,hlu,hlx⟩ := capEdges_bounds hL
obtain ⟨hro,hri,hru,hrx⟩ := capEdges_bounds (x := -R) (by linarith)
obtain ⟨hao,hai,hau,hax⟩ := capEdges_bounds hA
obtain ⟨hloo,hlio,hluo,hlxo⟩ := capEdges_bounds hlo
obtain ⟨hroo,hrio,hruo,hrxo⟩ := capEdges_bounds hro
obtain ⟨haoo,haio,hauo,haxo⟩ := capEdges_bounds hao
unfold capOuterLeft capOuterLeftOne capOuterRightOne capOuterRight capOuterBottom capOuterBottomOne
refine ⟨hloo,hlio,hluo,?_,?_,?_,haoo,haio,?_,?_,?_⟩ <;> linarith
theorem capOuterRectangle_subset_domain {L R A b : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) (hb : b < 2) :
closedRectangle (capOuterLeft L) (capOuterRight R) (capOuterBottom A) b ⊆ capChartDomain := by
obtain ⟨hlo,hli,hlu,hri,hrr,hro,hao,hai,hau,hlx,hrx⟩ := capOuterRectangle_edges hL hR hA
intro p hp
exact ⟨⟨hlo.trans_le hp.1.1, hp.1.2.trans_lt hro⟩,
⟨hao.trans_le hp.2.1,hp.2.2.trans_lt hb⟩⟩
theorem capSpatialCutoff_outer_boundary {L R A b : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) :
(∀ s ∈ Icc (capOuterBottom A) b,
capSpatialCutoff L R A (boxPoint (capOuterLeft L) s) = 0 ∧
capSpatialCutoff L R A (boxPoint (capOuterRight R) s) = 0) ∧
(∀ t ∈ Icc (capOuterLeft L) (capOuterRight R),
capSpatialCutoff L R A (boxPoint t (capOuterBottom A)) = 0) := by
obtain ⟨hlo,hli,hlu,hlx⟩ := capEdges_bounds hL
obtain ⟨hro,hri,hru,hrx⟩ := capEdges_bounds (x := -R) (by linarith)
obtain ⟨hao,hai,hau,hax⟩ := capEdges_bounds hA
have hlout : capOuterLeft L < capOuterEdge L := by
have hh := capEdges_bounds hlo
exact hh.2.1.trans hh.2.2.2
have hrout : -capOuterEdge (-R) < capOuterRight R := by
have hh := capEdges_bounds hro
exact neg_lt_neg (hh.2.1.trans hh.2.2.2)
have haout : capOuterBottom A < capOuterEdge A := by
have hh := capEdges_bounds hao
exact hh.2.1.trans hh.2.2.2
constructor
· intro s hs
constructor
· apply threeEdgeCutoff_zero hli (neg_lt_neg hri) hai
exact Or.inl (by simpa [boxPoint] using hlout.le)
· apply threeEdgeCutoff_zero hli (neg_lt_neg hri) hai
exact Or.inr (Or.inl (by simpa [boxPoint] using hrout.le))
· intro t ht
apply threeEdgeCutoff_zero hli (neg_lt_neg hri) hai
exact Or.inr (Or.inr (by simpa [boxPoint] using haout.le))
end SmoothLocal.Weighted
end
end OAI
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