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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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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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