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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 | import OAI.Geometry.CylinderCovering.Caps.CapsDefinitions
import OAI.Geometry.CylinderCovering.Radial.CapCoverage
namespace OAI
open Set
noncomputable section
namespace CylinderCovering.HalfHeightCaps
theorem covered_of_padding {δ : ℝ} {R b : ℝ → ℝ}
(hδ : 0 < δ)
(hcap : ∀ r ∈ Icc (-1 : ℝ) 1,
2 * R r = 1 + (δ / 2) ^ 2 * (RadialCaps.H r + b r))
(hb : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ b r)
(hpadding : ∀ p ∈ Icc (-1 : ℝ) 1, ∀ q ∈ Icc (-1 : ℝ) 1,
9 * (δ / 2) ≤ b p + b q) : Covered δ R := by
intro X Y Z hX hY
have hx : |2 * Y| ≤ 1 + (2 * Z - 1) := by
rw [abs_mul]
norm_num
linarith only [hY]
have hy : |2 * X| ≤ 1 - (2 * Z - 1) := by
rw [abs_mul]
norm_num
linarith only [hX]
have h := RadialCapCoverage.padding_coverage (eps := δ / 2)
(R := fun r => 2 * R r) (b := b) (by positivity) hcap hb hpadding hx hy
rcases h with ⟨q, ρ, hq, hρ, hR, heq, ht⟩ | ⟨p, σ, hp, hσ, hR, heq, ht⟩
· right
refine ⟨q, ρ / 2, hq, by positivity, by linarith only [hR], ?_, ?_⟩
· unfold alpha
unfold RadialCaps.G at heq
nlinarith only [heq]
· unfold beta
nlinarith only [ht]
· left
refine ⟨p, σ / 2, hp, by positivity, by linarith only [hR], ?_, ?_⟩
· unfold alpha
unfold RadialCaps.G at heq
nlinarith only [heq]
· unfold beta
nlinarith only [ht]
theorem cubic_dictionary (δ r : ℝ) :
2 * cubicCap δ r = RadialCaps.cubicCap 64 (δ / 2) r := by
unfold cubicCap F RadialCaps.cubicCap RadialCaps.H
ring
theorem cubic_covers {δ : ℝ} (hδ : 0 < δ) : Covered δ (cubicCap δ) := by
apply covered_of_padding (b := fun _ => 32 * δ) hδ
· intro r hr
unfold cubicCap F RadialCaps.H
ring
· intro r hr
positivity
· intro p hp q hq
linarith only [hδ]
theorem fractional_covers {δ : ℝ} (hδ : 0 < δ) (hsmall : δ ≤ 1) :
Covered δ (fractionalCap δ) := by
have hs : 0 ≤ Real.sqrt δ := Real.sqrt_nonneg δ
have hs1 : Real.sqrt δ ≤ 1 := by nlinarith [Real.sq_sqrt hδ.le]
have hδs : δ ≤ Real.sqrt δ := by nlinarith [Real.sq_sqrt hδ.le]
apply covered_of_padding (b := fun _ => 8 * Real.sqrt δ) hδ
· intro r hr
unfold fractionalCap F RadialCaps.H
rw [RadialCapCoverage.fractional_power δ hδ]
ring
· intro r hr
positivity
· intro p hp q hq
nlinarith only [hδs, hs]
theorem fractional_dictionary {δ : ℝ} (hδ : 0 < δ) (r : ℝ) :
2 * fractionalCap δ r = RadialCaps.fractionalCap (8 * Real.sqrt 2) (δ / 2) r := by
unfold fractionalCap RadialCaps.fractionalCap F RadialCaps.H
rw [RadialCapCoverage.fractional_power δ hδ,
RadialCapCoverage.fractional_power (δ / 2) (by positivity), Real.sqrt_div hδ.le]
have hs2 : (Real.sqrt 2) ^ 2 = 2 := Real.sq_sqrt (by norm_num)
have hs0 : Real.sqrt 2 ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (by norm_num))
field_simp
nlinarith only [hs2]
theorem halfheight_coverage : CoverageStatement := by
refine ⟨fun δ hδ => cubic_covers hδ, ⟨1, by norm_num, ?_⟩,
cubic_dictionary, ?_⟩
· exact fun δ hδ hs => fractional_covers hδ hs
· exact fun δ r hδ => fractional_dictionary hδ r
theorem compensation : CompensationStatement := by
intro δ X Y Z r s u v hY hZ hX hZv
unfold alpha beta at *
linear_combination δ * X * hY - δ * Y * hX - hZ - hZv
theorem physical_covers_of_covered {δ : ℝ} {R : ℝ → ℝ} (h : Covered δ R) :
physicalTetrahedron ⊆ firstPhysicalSweep δ R ∪ secondPhysicalSweep δ R := by
rintro z ⟨X, Y, Z, hX, hY, rfl⟩
have hX1 : |X| ≤ 1 := by linarith only [hX, hY, abs_nonneg Y]
have hY1 : |Y| ≤ 1 := by linarith only [hX, hY, abs_nonneg X]
rcases h X Y Z hX hY with ⟨r, u, hr, hu, hR, hYr, hZr⟩ |
⟨s, v, hs, hv, hR, hXs, hZs⟩
· left
refine ⟨r, abs_le.mp hr, u, ⟨hu, hR⟩, X, hX1, ?_⟩
rw [← hYr, ← hZr]
· right
refine ⟨s, abs_le.mp hs, v, ⟨hv, hR⟩, Y, hY1, ?_⟩
have hZs' : Z = 1 - v + δ * Y * beta s := by linarith only [hZs]
rw [← hXs, ← hZs']
theorem halfheight_physical_coverage : PhysicalCoverageStatement := by
constructor
· exact fun δ hδ => physical_covers_of_covered (cubic_covers hδ)
· exact fun δ hδ hs => physical_covers_of_covered (fractional_covers hδ hs)
theorem fractional_defect (a X Y : ℝ) :
X ^ 2 * (a / 2 + X ^ 2 / (2 * a)) + Y ^ 2 * (a / 2 + Y ^ 2 / (2 * a)) -
(a / 2 + X ^ 2 / (2 * a)) * Y ^ 2 -
(a / 2 + Y ^ 2 / (2 * a)) * X ^ 2 = (X ^ 2 - Y ^ 2) ^ 2 / (2 * a) := by
ring
end CylinderCovering.HalfHeightCaps
end
end OAI
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