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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | import OAI.Geometry.CylinderCovering.Physical.Paired
import OAI.Geometry.CylinderCovering.Radial.SqrtCap
import OAI.Geometry.CylinderCovering.Radial.CapCoverage
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace CylinderCovering.UnitEdgeFootprints
open TriangularCovering
lemma cap_dictionary (η q : ℝ) :
RadialSqrtCap.cap (1 / 120) (η / 2) q = 2 * sourceCap η (q / 2) := by
have hleft : 0 ≤ RadialSqrtCap.cap (1 / 120) (η / 2) q := Real.sqrt_nonneg _
have hright : 0 ≤ 2 * sourceCap η (q / 2) := mul_nonneg (by norm_num) (sourceCap_pos η (q / 2)).le
apply (sq_eq_sq₀ hleft hright).mp
rw [RadialSqrtCap.cap_sq (by norm_num : (0 : ℝ) ≤ 1 / 120), mul_pow]
have hs : (sourceCap η (q / 2)) ^ 2 =
1 / 4 + η ^ 2 * ((q / 2) ^ 2 / 4 + (q / 2) ^ 4 + 1 / 960) := by
unfold sourceCap
exact Real.sq_sqrt (by positivity)
rw [hs]
unfold RadialSqrtCap.profile RadialCaps.H
ring
theorem source_coverage : CoverageStatement := by
obtain ⟨δ, hδ, hpad⟩ := RadialSqrtCap.exists_padding_threshold
(by norm_num : (0 : ℝ) < 1 / 120)
refine ⟨2 * δ, by positivity, ?_⟩
intro η hη hsmall z hz
obtain ⟨t, ht, hz2, hx, hy⟩ := hz
have he : 0 < η / 2 := by positivity
obtain ⟨hεβ, hcap⟩ := hpad (η / 2) he (by linarith)
have hx' : |4 * z 1| ≤ 1 + (2 * t - 1) := by
rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 4)]
linarith
have hy' : |4 * z 0| ≤ 1 - (2 * t - 1) := by
rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 4)]
linarith
have hc := RadialCapCoverage.padding_coverage (eps := η / 2)
(R := RadialCaps.linearCap (1 / 240) (η / 2)) (b := fun _ => (1 / 240 : ℝ)) he
(fun _ _ => rfl) (fun _ _ => by norm_num)
(fun _ _ _ _ => by linarith) hx' hy'
rcases hc with hu | hl
· obtain ⟨q, ρ, hq, hρ, hR, hrep, htrep⟩ := hu
right
have hqI := abs_le.mp hq
have hcapq := hcap q hq
have hρcap : ρ ≤ 2 * sourceCap η (q / 2) := by
rw [← cap_dictionary]
apply hR.trans
simpa only [RadialCaps.linearCap, show (1 / 120 : ℝ) / 2 = 1 / 240 by norm_num] using hcapq
refine ⟨q / 2, ⟨by linarith, by linarith⟩, ρ / 2,
⟨by positivity, by linarith⟩, z 1, ?_, ?_⟩
· linarith [ht.2]
· ext i
fin_cases i
· change z 0 = q / 2 * (ρ / 2) - η * z 1 * (1 / 4 + (q / 2) ^ 2)
unfold RadialCaps.G at hrep
nlinarith only [hrep]
· rfl
· change z 2 = height * (1 - ρ / 2 + 2 * η * z 1 * (q / 2))
rw [hz2]
congr 1
nlinarith only [htrep]
· obtain ⟨q, ρ, hq, hρ, hR, hrep, htrep⟩ := hl
left
have hqI := abs_le.mp hq
have hcapq := hcap q hq
have hρcap : ρ ≤ 2 * sourceCap η (q / 2) := by
rw [← cap_dictionary]
apply hR.trans
simpa only [RadialCaps.linearCap, show (1 / 120 : ℝ) / 2 = 1 / 240 by norm_num] using hcapq
refine ⟨q / 2, ⟨by linarith, by linarith⟩, ρ / 2,
⟨by positivity, by linarith⟩, z 0, ?_, ?_⟩
· linarith [ht.1]
· ext i
fin_cases i
· rfl
· change z 1 = q / 2 * (ρ / 2) + η * z 0 * (1 / 4 + (q / 2) ^ 2)
unfold RadialCaps.G at hrep
nlinarith only [hrep]
· change z 2 = height * (ρ / 2 + 2 * η * z 0 * (q / 2))
rw [hz2]
congr 1
nlinarith only [htrep]
theorem finite_strict_of_coverage (hcoverage : CoverageStatement) : FiniteStrictStatement := by
obtain ⟨δ, hδ, hcover⟩ := hcoverage
refine ⟨min δ (1 / 40), by positivity, ?_⟩
intro η hη hsmall
have hc := hcover η hη (lt_of_lt_of_le hsmall (min_le_left _ _))
have hs := source_cost_strict hη (le_trans hsmall.le (min_le_right _ _))
obtain ⟨C₁, C₂, hspec, hpair, hlim⟩ := paired_family_realization η hη
have hevent : ∀ᶠ N in atTop, totalArea (C₁ N) + totalArea (C₂ N) < height / 4 :=
hlim.eventually (gt_mem_nhds hs)
obtain ⟨N, hcost⟩ := (eventually_atTop.1 hevent)
let M := max N 1
have hM : 0 < M := lt_of_lt_of_le (by norm_num : 0 < 1) (le_max_right _ _)
refine ⟨M, hM, C₁ M, C₂ M, ?_, ?_, hcost M (le_max_left _ _)⟩
· intro i
obtain ⟨h1, h2, h3, h4, _, _⟩ := hspec M i
exact ⟨h1, h2, h3, h4⟩
· intro z hz
exact hpair M hM z (hc hz)
lemma tetrahedron_similarity : tetrahedron =
Similarity.map (0 : Space) (1 / 2) (LinearIsometryEquiv.refl ℝ Space) '' K := by
have hh : H / 2 = height := by
unfold H height
have hs : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num)
have hn : Real.sqrt 2 ≠ 0 := by positivity
field_simp
nlinarith only [hs]
ext z
constructor
· rintro ⟨t, ht, hz, hx, hy⟩
refine ⟨point (2 * z 0) (2 * z 1) t, ?_, ?_⟩
· refine ⟨2 * z 0, 2 * z 1, t, ht.1, ht.2, ?_, ?_, rfl⟩
· rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)]; linarith
· rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)]; linarith
· ext i
fin_cases i
· simp [Similarity.map, point]
· simp [Similarity.map, point]
· change 0 + (1 / 2 : ℝ) * (H * t) = z 2
rw [hz, ← hh]
ring
· rintro ⟨_, ⟨x, y, t, ht0, ht1, hx, hy, rfl⟩, rfl⟩
refine ⟨t, ⟨ht0, ht1⟩, ?_, ?_, ?_⟩
· change 0 + (1 / 2 : ℝ) * (H * t) = height * t
rw [← hh]
ring
· change |0 + (1 / 2 : ℝ) * x| ≤ (1 - t) / 2
rw [zero_add, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)]
linarith
· change |0 + (1 / 2 : ℝ) * y| ≤ t / 2
rw [zero_add, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)]
linarith
theorem minimum_area : Similarity.minimumArea tetrahedron = height / 2 := by
rw [tetrahedron_similarity, Similarity.minimumArea_map _ _
(by norm_num : (1 / 2 : ℝ) ≠ 0) (LinearIsometryEquiv.refl ℝ Space)]
change (1 / 2 : ℝ) ^ 2 * A_min = height / 2
rw [projection_minimum]
unfold H height
have hs : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num)
have hn : Real.sqrt 2 ≠ 0 := by positivity
field_simp
nlinarith only [hs]
theorem full_unit_edge : FullStatement :=
⟨local_realization, family_realization, paired_family_realization, source_cost,
source_coverage, finite_strict_of_coverage source_coverage, minimum_area⟩
end CylinderCovering.UnitEdgeFootprints
end
end OAI
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