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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Geometry.CylinderCovering.Singular.EdgeOneNormalizationDefinitions
import OAI.Geometry.CylinderCovering.Rational.Normalization
import OAI.Geometry.CylinderCovering.Ruled.Similarity
namespace OAI
noncomputable section
open Set MeasureTheory
namespace CylinderCovering.SingularEdgeOneCover
open RuledApproximation
lemma height_pos : 0 < height := by
unfold height
positivity
theorem tetrahedron_eq_centered_translation :
tetrahedron =
(fun z : Space => z + vec3 0 0 (height / 2)) '' SlopeRational.referenceTetrahedron := by
ext z
constructor
· rintro ⟨s, hs0, hs1, hx, hy, hz⟩
refine ⟨vec3 (z 0) (z 1) (z 2 - height / 2), ?_, ?_⟩
· refine ⟨s - 1 / 2, by linarith, by linarith, ?_, ?_, ?_⟩
· change |z 0| ≤ (1 / 2 + (s - 1 / 2)) / 2
simpa only [show (1 / 2 : ℝ) + (s - 1 / 2) = s by ring] using hx
· change |z 1| ≤ (1 / 2 - (s - 1 / 2)) / 2
simpa only [show (1 / 2 : ℝ) - (s - 1 / 2) = 1 - s by ring] using hy
· change z 2 - height / 2 = height * (s - 1 / 2)
rw [hz]
ring
· ext i
fin_cases i <;> simp [vec3]
· rintro ⟨w, ⟨t, ht0, ht1, hx, hy, hz⟩, rfl⟩
refine ⟨t + 1 / 2, by linarith, by linarith, ?_, ?_, ?_⟩
· change |w 0 + 0| ≤ (t + 1 / 2) / 2
simpa [add_comm] using hx
· change |w 1 + 0| ≤ (1 - (t + 1 / 2)) / 2
simpa only [add_zero, show (1 : ℝ) - (t + 1 / 2) = 1 / 2 - t by ring] using hy
· change w 2 + height / 2 = height * (t + 1 / 2)
change w 2 = height * t at hz
rw [hz]
ring
theorem shadowArea_eq_centered (u : Space) :
shadowArea tetrahedron u = shadowArea SlopeRational.referenceTetrahedron u := by
rw [tetrahedron_eq_centered_translation]
have hmap : (fun z : Space => z + vec3 0 0 (height / 2)) =
Similarity.map (vec3 0 0 (height / 2)) 1 (LinearIsometryEquiv.refl ℝ Space) := by
funext z
simp [Similarity.map, add_comm]
rw [hmap]
change Similarity.shadowArea
(Similarity.map (vec3 0 0 (height / 2)) 1 (LinearIsometryEquiv.refl ℝ Space) ''
SlopeRational.referenceTetrahedron) ((LinearIsometryEquiv.refl ℝ Space) u) = _
rw [Similarity.shadowArea_map _ _ (by norm_num)]
simp only [one_pow, one_mul]
rfl
theorem tetrahedron_minimumShadowArea : minimumShadowArea tetrahedron = height / 2 := by
have heq : minimumShadowArea tetrahedron =
minimumShadowArea SlopeRational.referenceTetrahedron := by
unfold minimumShadowArea
simp only [shadowArea_eq_centered]
rw [heq, SlopeRational.referenceTetrahedron_minimumShadowArea]
rfl
lemma minimumShadowArea_pos : 0 < minimumShadowArea tetrahedron := by
rw [tetrahedron_minimumShadowArea]
exact div_pos height_pos (by norm_num)
theorem same_family_half_bounds {n m : ℕ} (C : Fin n → Cylinder)
(D : Fin m → Cylinder) (B₁ B₂ : Cylinder)
(hcost : familyArea C + familyArea D + B₁.area + B₂.area < height / 4) :
familyArea C + familyArea D + B₁.area + B₂.area <
minimumShadowArea tetrahedron / 2 ∧
relativeCost tetrahedron C + relativeCost tetrahedron D +
B₁.area / shadowArea tetrahedron B₁.axis +
B₂.area / shadowArea tetrahedron B₂.axis < 1 / 2 := by
have hmin := minimumShadowArea_pos
have htotal : familyArea C + familyArea D + B₁.area + B₂.area <
minimumShadowArea tetrahedron / 2 := by
rw [tetrahedron_minimumShadowArea]
linarith only [hcost]
refine ⟨htotal, ?_⟩
have hC : relativeCost tetrahedron C ≤ familyArea C / minimumShadowArea tetrahedron :=
relativeCost_le_familyArea_div_minimum tetrahedron hmin C
have hD : relativeCost tetrahedron D ≤ familyArea D / minimumShadowArea tetrahedron :=
relativeCost_le_familyArea_div_minimum tetrahedron hmin D
have hB (B : Cylinder) : B.area / shadowArea tetrahedron B.axis ≤
B.area / minimumShadowArea tetrahedron :=
div_le_div_of_nonneg_left ENNReal.toReal_nonneg hmin
(Similarity.minimumArea_le_shadowArea tetrahedron B.unit_axis)
have hsum : relativeCost tetrahedron C + relativeCost tetrahedron D +
B₁.area / shadowArea tetrahedron B₁.axis +
B₂.area / shadowArea tetrahedron B₂.axis ≤
(familyArea C + familyArea D + B₁.area + B₂.area) /
minimumShadowArea tetrahedron := by
rw [add_div, add_div, add_div]
linarith only [hC, hD, hB B₁, hB B₂]
apply hsum.trans_lt
apply (div_lt_iff₀ hmin).mpr
linarith only [htotal]
theorem normalization : NormalizationStatement :=
⟨tetrahedron_eq_centered_translation, tetrahedron_minimumShadowArea,
fun _ _ C D B₁ B₂ hcost => same_family_half_bounds C D B₁ B₂ hcost⟩
end CylinderCovering.SingularEdgeOneCover
end
end OAI
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