| import OAI.Geometry.CylinderCovering.Angular.GeneralDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.GeneralCapCoverage |
| import OAI.Geometry.CylinderCovering.Convexity.GeneralCapArea |
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|
| namespace OAI |
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|
| noncomputable section |
| open Set MeasureTheory |
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|
| namespace CylinderCovering.AngularGeneral |
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|
| 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 |
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|
| 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 |
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| end |
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| end OAI |
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