| import OAI.Geometry.TriangularCovering.Main |
| import OAI.Geometry.CylinderCovering.Angular.FamilyDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| namespace TriangularCovering.AngularFamily |
|
|
| variable {n : ℕ} |
|
|
| def family (ε : ℝ) (a b A B R : Fin n → ℝ) |
| (hab : ∀ j, a j < b j) (hR : ∀ j, 0 < R j) : Fin (2 * n) → TriCylinder := |
| (Fin.append |
| (fun j => lowerCylinder (hab j) (hR j) ε (A j) (B j)) |
| (fun j => upperCylinder (hab j) (hR j) ε (A j) (B j))) ∘ |
| Fin.cast (two_mul n) |
|
|
| theorem total_area (ε : ℝ) (a b A B R : Fin n → ℝ) |
| (hab : ∀ j, a j < b j) (hR : ∀ j, 0 < R j) : |
| totalArea (family ε a b A B R hab hR) / H = |
| ∑ j : Fin n, (b j - a j) * (R j)^2 / |
| Real.sqrt (1 + ε^2 * ((A j)^2 + 2 * (B j)^2)) := by |
| have hs : totalArea (family ε a b A B R hab hR) = |
| (∑ j : Fin n, area (lowerCylinder (hab j) (hR j) ε (A j) (B j)).base) + |
| (∑ j : Fin n, area (upperCylinder (hab j) (hR j) ε (A j) (B j)).base) := by |
| change (∑ i : Fin (2 * n), |
| (fun k : Fin (n + n) => area (Fin.append |
| (fun j => lowerCylinder (hab j) (hR j) ε (A j) (B j)) |
| (fun j => upperCylinder (hab j) (hR j) ε (A j) (B j)) k).base) |
| ((finCongr (two_mul n)) i)) = _ |
| rw [Equiv.sum_comp (finCongr (two_mul n)) |
| (fun k => area (Fin.append |
| (fun j => lowerCylinder (hab j) (hR j) ε (A j) (B j)) |
| (fun j => upperCylinder (hab j) (hR j) ε (A j) (B j)) k).base)] |
| rw [Fin.sum_univ_add] |
| simp only [Fin.append_left, Fin.append_right] |
| rw [hs, ← Finset.sum_add_distrib, Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [lowerCylinder_area, upperCylinder_area] |
| field_simp [H_ne_zero] |
| ring |
|
|
| theorem family_prescribed (ε : ℝ) (a b A B R : Fin n → ℝ) |
| (hab : ∀ j, a j < b j) (hR : ∀ j, 0 < R j) : |
| PrescribedFamily (family ε a b A B R hab hR) ε a b A B R := by |
| constructor |
| · intro j |
| simp only [family,Function.comp_apply,Fin.cast_cast,Fin.cast_eq_self,Fin.append_left] |
| exact ⟨rfl,rfl⟩ |
| · intro j |
| simp only [family,Function.comp_apply,Fin.cast_cast,Fin.cast_eq_self,Fin.append_right] |
| exact ⟨rfl,rfl⟩ |
|
|
| theorem covers_of_coordinate_hits (ε : ℝ) (a b A B R : Fin n → ℝ) |
| (hab : ∀ j, a j < b j) (hR : ∀ j, 0 < R j) |
| (hcover : ∀ x y t : ℝ, |x| ≤ 1 - t → |y| ≤ t → |
| (∃ j, LowerHit ε (a j) (b j) (A j) (B j) (R j) x y t) ∨ |
| (∃ j, UpperHit ε (a j) (b j) (A j) (B j) (R j) x y t)) : |
| Covers (family ε a b A B R hab hR) := by |
| rintro z ⟨x, y, t, _ht0, _ht1, hx, hy, rfl⟩ |
| rcases hcover x y t hx hy with ⟨j, hj⟩ | ⟨j, hj⟩ |
| · refine ⟨Fin.cast (two_mul n).symm (Fin.castAdd n j), ?_⟩ |
| simp only [family, Function.comp_apply, Fin.cast_cast, Fin.cast_eq_self, |
| Fin.append_left] |
| exact lowerCylinder_hit _ _ _ _ _ _ _ _ hj.1 hj.2.1 hj.2.2.1 hj.2.2.2 |
| · refine ⟨Fin.cast (two_mul n).symm (Fin.natAdd n j), ?_⟩ |
| simp only [family, Function.comp_apply, Fin.cast_cast, Fin.cast_eq_self, |
| Fin.append_right] |
| exact upperCylinder_hit _ _ _ _ _ _ _ _ hj.1 hj.2.1 hj.2.2.1 hj.2.2.2 |
|
|
| theorem strict_bounds (ε : ℝ) (a b A B R : Fin n → ℝ) |
| (hab : ∀ j, a j < b j) (hR : ∀ j, 0 < R j) |
| (hcost : (∑ j : Fin n, (b j - a j) * (R j)^2 / |
| Real.sqrt (1 + ε^2 * ((A j)^2 + 2 * (B j)^2))) < 1 / 2) : |
| totalArea (family ε a b A B R hab hR) < A_min / 2 ∧ |
| relativeCost (family ε a b A B R hab hR) < 1 / 2 := by |
| have hratio : totalArea (family ε a b A B R hab hR) / H < 1/2 := by |
| rw [total_area] |
| exact hcost |
| constructor |
| · rw [projection_minimum] |
| have h := (div_lt_iff₀ H_pos).mp hratio |
| linarith only [h] |
| · exact (relativeCost_le_totalArea _).trans_lt hratio |
|
|
| end TriangularCovering.AngularFamily |
|
|
| end |
|
|
| end OAI |
|
|
