| import OAI.Geometry.CylinderCovering.Cubic.RescalingDefinitions |
| import OAI.Geometry.CylinderCovering.Cubic.Rescaling |
| import OAI.Geometry.CylinderCovering.Angular.CubicFamily |
| import OAI.Geometry.CylinderCovering.Convexity.CylinderSimilarity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
|
|
| namespace CylinderCovering.Cubic |
| open TriangularCovering |
|
|
| private abbrev identityIsometry : Space ≃ₗᵢ[ℝ] Space := LinearIsometryEquiv.refl ℝ Space |
|
|
| lemma contraction_point (x y z : ℝ) : |
| contraction (point x y z) = point (x / 2) (y / 2) ((z + 1) / 2) := by |
| ext i |
| fin_cases i <;> dsimp [contraction, point] <;> ring |
|
|
| lemma contraction_double_sub (p : Space) : |
| contraction ((2 : ℝ) • p - point 0 0 1) = p := by |
| have hs : point 0 0 (1 / 2) = (1 / 2 : ℝ) • point 0 0 1 := by |
| ext i |
| fin_cases i <;> dsimp [point] <;> ring |
| rw [contraction, hs] |
| module |
|
|
| lemma contraction_K4 : contraction '' K4 = K := by |
| rw [K4_eq_double_sub, Set.image_image] |
| have hf : (fun p : Space => contraction ((2 : ℝ) • p - point 0 0 1)) = id := by |
| funext p |
| exact contraction_double_sub p |
| rw [hf, Set.image_id] |
|
|
| def contractCylinder (c : TriCylinder) : TriCylinder where |
| axis := c.axis |
| base := contractedBaseMap c.axis '' c.base |
| unit_axis := c.unit_axis |
| perpendicular := by |
| exact Similarity.transportedBase_perpendicular c.unit_axis c.perpendicular |
| (point 0 0 (1 / 2)) (1 / 2) identityIsometry |
| triangular := Similarity.transportedBase_triangular c.triangular c.axis |
| (point 0 0 (1 / 2)) (by norm_num : (1 / 2 : ℝ) ≠ 0) identityIsometry |
| compact := Similarity.transportedBase_compact c.compact c.axis |
| (point 0 0 (1 / 2)) (1 / 2) identityIsometry |
| convex := Similarity.transportedBase_convex c.convex c.axis |
| (point 0 0 (1 / 2)) (1 / 2) identityIsometry |
| measurable := (Similarity.transportedBase_compact c.compact c.axis |
| (point 0 0 (1 / 2)) (1 / 2) identityIsometry).measurableSet |
| finite_area := Similarity.transportedBase_finite c.finite_area c.axis |
| (point 0 0 (1 / 2)) (by norm_num : (1 / 2 : ℝ) ≠ 0) identityIsometry |
|
|
| lemma contractCylinder_axis (c : TriCylinder) : (contractCylinder c).axis = c.axis := rfl |
|
|
| lemma contractCylinder_carrier (c : TriCylinder) : |
| contraction '' c.carrier = (contractCylinder c).carrier := by |
| exact Similarity.carrier_image c.base c.axis (point 0 0 (1 / 2)) |
| (by norm_num : (1 / 2 : ℝ) ≠ 0) identityIsometry |
|
|
| lemma contractCylinder_area (c : TriCylinder) : |
| area (contractCylinder c).base = area c.base / 4 := by |
| change Similarity.area (Similarity.transportedBase c.base c.axis |
| (point 0 0 (1 / 2)) (1 / 2) identityIsometry) = Similarity.area c.base / 4 |
| rw [Similarity.area_transportedBase c.base c.axis (point 0 0 (1 / 2)) |
| (by norm_num : (1 / 2 : ℝ) ≠ 0) identityIsometry] |
| ring |
|
|
| lemma contractFamily_covers {m : ℕ} {c : Fin m → TriCylinder} (hc : CoversK4 c) : |
| Covers (fun i => contractCylinder (c i)) := by |
| intro p hp |
| rw [← contraction_K4] at hp |
| obtain ⟨q, hq, rfl⟩ := hp |
| obtain ⟨i, hi⟩ := hc q hq |
| refine ⟨i, ?_⟩ |
| rw [← contractCylinder_carrier] |
| exact ⟨q, hi, rfl⟩ |
|
|
| lemma contractFamily_total {m : ℕ} (c : Fin m → TriCylinder) : |
| totalArea (fun i => contractCylinder (c i)) = totalArea c / 4 := by |
| simp only [totalArea, contractCylinder_area, Finset.sum_div] |
|
|
| theorem rescaledFiniteCover_proof : RescaledFiniteCoverStatement := by |
| intro τ hτ hτ1 |
| refine ⟨rescaled_count τ, ?_⟩ |
| obtain ⟨c, hc, harea, hprescribed⟩ := finiteCover_proof (τ / 2) |
| (by linarith) (by linarith) |
| refine ⟨fun i => contractCylinder (c i), contractFamily_covers hc, ?_, ?_⟩ |
| · rw [contractFamily_total, harea] |
| · exact ⟨c, hprescribed, fun i => ⟨rfl, rfl⟩⟩ |
|
|
| end CylinderCovering.Cubic |
|
|
| end |
|
|
| end OAI |
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|
