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785c727 | 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 | import OAI.Geometry.CylinderCovering.Angular.GeneralDefinitions
import OAI.Geometry.CylinderCovering.Convexity.GeneralCapCoverage
import OAI.Geometry.CylinderCovering.Convexity.GeneralCapArea
namespace OAI
noncomputable section
open Set MeasureTheory
namespace CylinderCovering.AngularGeneral
theorem coordinate_cover {τ η E : ℝ} {D : Data}
(hD : Admissible τ η E D) (hτ : 0 < τ) (hτ1 : τ ≤ 1/2)
(hE : 0 ≤ E) (hη : 0 < η) (hη1 : η ≤ 1/480)
(hsmall : (10+4*E)*τ ≤ η)
{x y t : ℝ} (hx : |x| ≤ 1-t) (hy : |y| ≤ t) :
(∃ j < D.count, ∃ q ∈ interval D j,
0 ≤ t-τ*x*D.beta j ∧ t-τ*x*D.beta j ≤ D.radius j q ∧
y = (t-τ*x*D.beta j)*q+τ*x*D.alpha j) ∨
(∃ j < D.count, ∃ q ∈ interval D j,
0 ≤ 1-t+τ*y*D.beta j ∧ 1-t+τ*y*D.beta j ≤ D.radius j q ∧
x = (1-t+τ*y*D.beta j)*q-τ*y*D.alpha j) := by
rcases hD with ⟨hn,hnode0,hnoden,hstep,hbound,hL,hcont,hpositive,herror⟩
obtain ⟨L,hLc,hL0,hL1,hcompat⟩ := hL
let boundary := fun j => L (D.node j)
have hb0 : boundary 0 = 0 := by simp [boundary,hnode0,hL0]
have hbn : boundary D.count = 0 := by simp [boundary,hnoden,hL1]
have hleft : ∀ j < D.count, D.alpha j-D.node j*D.beta j = boundary j := by
intro j hj
exact hcompat j hj (D.node j) ⟨le_rfl,(hstep j hj).le⟩
have hright : ∀ j < D.count, D.alpha j-D.node (j+1)*D.beta j = boundary (j+1) := by
intro j hj
exact hcompat j hj (D.node (j+1)) ⟨(hstep j hj).le,le_rfl⟩
apply GeneralCap.coordinate_coverage hn D.node D.alpha D.beta boundary D.radius
hnode0 hnoden hbound hb0 hbn hstep hleft hright hτ hτ1 hE hη hη1 hsmall
· intro j hj q hq
exact ⟨(herror j hj q hq).1,(herror j hj q hq).2.1⟩
· intro j hj q hq
have hh := (abs_le.mp (herror j hj q hq).2.2).1
dsimp [profile] at hh
linarith only [hh]
· exact hx
· exact hy
theorem integralCost_expansion {τ η E : ℝ} {D : Data}
(hD : Admissible τ η E D) (hη : 0 ≤ η) (hη1 : η ≤ 1/8)
(hE : 0 ≤ E) (hτ : τ^2 ≤ 1) (hsmall : E*τ^2 ≤ 1/16) :
|integralCost τ D-(1/2+(2*η-1/240)*τ^2)| ≤ 2*(3*E+1)*τ^4 := by
rcases hD with ⟨hn,hnode0,hnoden,hstep,hbound,hL,hcont,hpositive,herror⟩
have hh := GeneralCapArea.general_cap_area_expansion hnode0 hnoden
(fun j hj => (hstep j hj).le) hcont hη hη1 hE hτ hsmall
(fun j hj q hq => (herror j hj q hq).1)
(fun j hj q hq => (herror j hj q hq).2.1)
(fun j hj q hq => (herror j hj q hq).2.2)
rw [← Fin.sum_univ_eq_sum_range (fun j : ℕ =>
∫ q in D.node j..D.node (j+1),
GeneralCapArea.capDensity τ (D.alpha j) (D.beta j) (D.radius j) q) D.count] at hh
exact hh
end CylinderCovering.AngularGeneral
end
end OAI
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