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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 | import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions
import OAI.Geometry.CylinderCovering.Rational.RationalThreshold
namespace OAI
noncomputable section
open Set Filter
open scoped Topology
namespace SlopeRational
theorem rational_cover_small {d : ℝ} (hd : 0 < d) :
∃ δ C : ℝ, 0 < δ ∧ 0 < C ∧ ∀ ε q : ℝ,
0 < ε → ε < δ → C * ε ^ 3 ≤ q →
referenceTetrahedron ⊆ lowerSweep d ε q ∪ upperSweep d ε q := by
obtain ⟨δ, C, hδ, hC, hinc⟩ := thresholdInclusion_main d hd
refine ⟨δ, C, hδ, hC, ?_⟩
intro ε q he hsmall hq z hz
obtain ⟨t, ht0, ht1, hx, hy, hz2⟩ := hz
have hx1 : |z 0| ≤ 1 / 2 := by linarith
have hy1 : |z 1| ≤ 1 / 2 := by linarith
have ht : |t| ≤ 1 / 2 := abs_le.mpr ⟨ht0, ht1⟩
have hwy : vec2 (z 1) t ∈ wedge := by
change 2 * |z 1| ≤ 1 / 2 - t
linarith
have hwx : vec2 (z 0) (-t) ∈ wedge := by
change 2 * |z 0| ≤ 1 / 2 - -t
linarith
rcases threshold_alternative hd ε (z 0) (z 1) t with hfirst | hsecond
· obtain ⟨p, hp, hmap⟩ := hinc ε q (z 0) (z 1) t he hsmall hq hx1 hy1 ht hwy hfirst
apply Or.inl
refine ⟨p, hp, z 0, hx1, ?_⟩
have hY := congrArg (fun w : Plane => w 0) hmap
have hT := congrArg (fun w : Plane => w 1) hmap
change p 0 + ε * z 0 * a d (p 0) (p 1) = z 1 at hY
change p 1 + ε * z 0 * b d (p 0) (p 1) = t at hT
ext i
fin_cases i
· rfl
· change z 1 = p 0 + z 0 * (ε * a d (p 0) (p 1))
nlinarith only [hY]
· change z 2 = physicalScale * (p 1 + z 0 * (ε * b d (p 0) (p 1)))
rw [hz2, ← hT]
ring
· obtain ⟨p, hp, hmap⟩ := hinc ε q (-(z 1)) (z 0) (-t) he hsmall hq
(by simpa only [abs_neg] using hy1) hx1 (by simpa only [abs_neg] using ht) hwx hsecond
apply Or.inr
let w : Space := RuledApproximation.vec3 (z 1)
(p 0 + z 1 * ((-ε • field d p) 0))
(physicalScale * (p 1 + z 1 * ((-ε • field d p) 1)))
refine ⟨w, ⟨p, hp, z 1, hy1, rfl⟩, ?_⟩
have hY := congrArg (fun v : Plane => v 0) hmap
have hT := congrArg (fun v : Plane => v 1) hmap
change p 0 + ε * (-(z 1)) * a d (p 0) (p 1) = z 0 at hY
change p 1 + ε * (-(z 1)) * b d (p 0) (p 1) = -t at hT
ext i
fin_cases i
· change p 0 + z 1 * (-ε * a d (p 0) (p 1)) = z 0
nlinarith only [hY]
· rfl
· change -(physicalScale * (p 1 + z 1 * (-ε * b d (p 0) (p 1)))) = z 2
rw [hz2]
have hid : p 1 + z 1 * (-ε * b d (p 0) (p 1)) = -t := by nlinarith only [hT]
rw [hid]
ring
theorem rationalCoverage_main : RationalCoverageStatement := by
intro d q hd hqpos _hqO hqsmall
obtain ⟨δ, C, hδ, hC, hcover⟩ := rational_cover_small hd
have hepos : ∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), 0 < ε := self_mem_nhdsWithin
have hesmall : ∀ᶠ ε : ℝ in 𝓝[>] (0 : ℝ), ε < δ :=
(eventually_lt_nhds hδ).filter_mono nhdsWithin_le_nhds
have hbound := hqsmall.def (show 0 < 1 / C by positivity)
filter_upwards [hepos, hesmall, hqpos, hbound] with ε he heδ hq hbound
have hb : ε ^ 3 ≤ (1 / C) * q ε := by
simpa only [Real.norm_eq_abs, abs_of_pos (pow_pos he 3), abs_of_pos hq] using hbound
have hpad : C * ε ^ 3 ≤ q ε := by
have hmul := mul_le_mul_of_nonneg_left hb hC.le
have hne : C ≠ 0 := ne_of_gt hC
field_simp [hne] at hmul
nlinarith only [hmul]
exact hcover ε (q ε) he heδ hpad
end SlopeRational
end
end OAI
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