| import OAI.Geometry.CylinderCovering.Angular.CubicCoverDefinitions |
| import OAI.Geometry.CylinderCovering.Angular.Margins |
| import OAI.Geometry.CylinderCovering.Cubic.LowerGeometry |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CylinderCovering.Cubic |
| open Set TriangularCovering |
|
|
| lemma family_node_strict {N : ℕ} (hN : 0 < N) (j : ℕ) : |
| node N j < node N (j + 1) := by |
| have := node_succ N j |
| have := step_pos hN |
| linarith |
|
|
| lemma family_cap_pos {ε : ℝ} (hε : 0 ≤ ε) (a b : ℝ) : 0 < cap ε a b := by |
| have ha : 0 ≤ profile a := by unfold profile; positivity |
| have hm : 0 ≤ profileMax a b := ha.trans (le_max_left _ _) |
| unfold cap |
| positivity |
|
|
| def lowerFamily4 (ε : ℝ) (hε : 0 < ε) (j : Fin (n ε)) : TriCylinder := |
| CubicLowerGeometry.lowerCylinder |
| (family_node_strict (ceil_count_pos hε) j) |
| (family_cap_pos hε.le (node (n ε) j) (node (n ε) (j + 1))) ε |
| (alpha (node (n ε) j) (node (n ε) (j + 1))) |
| (beta (node (n ε) j) (node (n ε) (j + 1))) |
|
|
| def upperFamily4 (ε : ℝ) (hε : 0 < ε) (j : Fin (n ε)) : TriCylinder := |
| TriangularCovering.upperCylinder |
| (family_node_strict (ceil_count_pos hε) j) |
| (family_cap_pos hε.le (node (n ε) j) (node (n ε) (j + 1))) ε |
| (alpha (node (n ε) j) (node (n ε) (j + 1))) |
| (beta (node (n ε) j) (node (n ε) (j + 1))) |
|
|
| def finiteFamily4 (ε : ℝ) (hε : 0 < ε) : Fin (2 * n ε) → TriCylinder := |
| (Fin.append (lowerFamily4 ε hε) (upperFamily4 ε hε)) ∘ Fin.cast (two_mul (n ε)) |
|
|
| lemma finiteFamily4_covers_of_coordinate_cover {ε : ℝ} (hε : 0 < ε) |
| (hcover : ∀ x y z : ℝ, |x| ≤ 1 - z → |y| ≤ 1 + z → |
| (∃ j : ℕ, j < n ε ∧ LowerHit (n ε) j ε x y z) ∨ |
| (∃ j : ℕ, j < n ε ∧ UpperHit (n ε) j ε x y z)) : |
| CoversK4 (finiteFamily4 ε hε) := by |
| rintro p ⟨x, y, z, _hz0, _hz1, hx, hy, rfl⟩ |
| rcases hcover x y z hx hy with ⟨j, hj, hl⟩ | ⟨j, hj, hu⟩ |
| · refine ⟨Fin.cast (two_mul (n ε)).symm |
| (Fin.castAdd (n ε) (⟨j, hj⟩ : Fin (n ε))), ?_⟩ |
| simp only [finiteFamily4, Function.comp_apply, Fin.cast_cast, |
| Fin.cast_eq_self, Fin.append_left] |
| exact CubicLowerGeometry.lowerCylinder_hit _ _ _ _ _ _ _ _ |
| hl.1.1 hl.2 hl.1.2.1 hl.1.2.2 |
| · refine ⟨Fin.cast (two_mul (n ε)).symm |
| (Fin.natAdd (n ε) (⟨j, hj⟩ : Fin (n ε))), ?_⟩ |
| simp only [finiteFamily4, Function.comp_apply, Fin.cast_cast, |
| Fin.cast_eq_self, Fin.append_right] |
| exact TriangularCovering.upperCylinder_hit _ _ _ _ _ _ _ _ |
| hu.1.1 hu.2 hu.1.2.1 hu.1.2.2 |
|
|
| lemma finiteFamily4_total (ε : ℝ) (hε : 0 < ε) : |
| totalArea (finiteFamily4 ε hε) = cubicCost ε := by |
| have hs : totalArea (finiteFamily4 ε hε) = |
| (∑ j : Fin (n ε), area (lowerFamily4 ε hε j).base) + |
| (∑ j : Fin (n ε), area (upperFamily4 ε hε j).base) := by |
| change (∑ i : Fin (2 * n ε), |
| (fun k : Fin (n ε + n ε) => |
| area (Fin.append (lowerFamily4 ε hε) (upperFamily4 ε hε) k).base) |
| ((finCongr (two_mul (n ε))) i)) = _ |
| rw [Equiv.sum_comp (finCongr (two_mul (n ε))) |
| (fun k => area (Fin.append (lowerFamily4 ε hε) (upperFamily4 ε hε) k).base)] |
| rw [Fin.sum_univ_add] |
| simp only [Fin.append_left, Fin.append_right] |
| rw [hs, ← Finset.sum_add_distrib, cubicCost, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [lowerFamily4, upperFamily4, CubicLowerGeometry.lowerCylinder_area, |
| TriangularCovering.upperCylinder_area, node_succ] |
| dsimp only [n] |
| ring |
|
|
| lemma finiteFamily4_prescribed (ε : ℝ) (hε : 0 < ε) : |
| PrescribedFamily ε (finiteFamily4 ε hε) := by |
| intro j |
| simp only [lowerIndex, upperIndex, finiteFamily4, Function.comp_apply, |
| Fin.cast_cast, Fin.cast_eq_self, Fin.append_left, Fin.append_right] |
| exact ⟨rfl, rfl, rfl, rfl⟩ |
|
|
| theorem finiteCover_proof : FiniteCoverStatement := by |
| intro ε hε hsmall |
| refine ⟨finiteFamily4 ε hε, ?_, finiteFamily4_total ε hε, |
| finiteFamily4_prescribed ε hε⟩ |
| apply finiteFamily4_covers_of_coordinate_cover hε |
| intro x y z hx hy |
| exact finite_coordinate_cover_ceil hε hsmall hx hy |
|
|
| end CylinderCovering.Cubic |
|
|
| end |
|
|
| end OAI |
|
|
