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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.Radial.CapVariantsDefinitions
import OAI.Geometry.CylinderCovering.Radial.CapCosts
import OAI.Geometry.CylinderCovering.Radial.CapCoverage
import OAI.Geometry.CylinderCovering.Radial.PairedSweep
namespace OAI
noncomputable section
open Set Filter
open scoped Topology
namespace CylinderCovering.RadialCapVariants
lemma physicalCost_eq_Q (ε : ℝ) (R : ℝ → ℝ) :
RadialPairedSweep.physicalCost (Real.sqrt 2) ε R = RadialCaps.Q R ε := by
unfold RadialPairedSweep.physicalCost RadialCaps.Q RadialCaps.D0
simp only [Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 2)]
theorem coverLimit_of_paired (hpaired : RadialPairedSweep.PairedStatement)
{ε : ℝ} {R : ℝ → ℝ}
(hR : ContinuousOn R (Icc (-1 : ℝ) 1))
(hR0 : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ R r) (hcover : SweepsCover ε R) :
CoverLimit ε R := by
intro tagU tagL htagU htagL
obtain ⟨U, L, hUL, hlim⟩ := hpaired (Real.sqrt 2) ε (by positivity) R hR hR0
tagU tagL htagU htagL
refine ⟨U, L, ?_, ?_⟩
· intro N hN
obtain ⟨haxis, hU, hL⟩ := hUL N hN
refine ⟨haxis, hU, hL, ?_⟩
intro z hz
rcases hcover hz with hzU | hzL
· exact Or.inl (hU z hzU)
· exact Or.inr (hL z hzL)
· simpa only [physicalCost_eq_Q] using hlim
theorem finiteSaving_of_coverLimit {ε : ℝ} {R : ℝ → ℝ}
(hcover : CoverLimit ε R) (hQ : RadialCaps.Q R ε < 1) : FiniteSaving ε := by
intro tagU tagL htagU htagL
obtain ⟨U, L, hUL, hlim⟩ := hcover tagU tagL htagU htagL
have hlt : 2 * Real.sqrt 2 * RadialCaps.Q R ε < 2 * Real.sqrt 2 := by
nlinarith [Real.sqrt_pos.2 (by norm_num : (0 : ℝ) < 2)]
have hcost : ∀ᶠ N in atTop, TriangularCovering.totalArea (U N) +
TriangularCovering.totalArea (L N) < 2 * Real.sqrt 2 :=
hlim.eventually (gt_mem_nhds hlt)
obtain ⟨N, hN, hcostN⟩ := ((Filter.eventually_gt_atTop (0 : ℕ)).and hcost).exists
obtain ⟨haxis, _, _, hcoverN⟩ := hUL N hN
exact ⟨N, hN, U N, L N, haxis, hcoverN, hcostN⟩
theorem assemble
(hpaired : RadialPairedSweep.PairedStatement)
(hlinear : ∀ β : ℝ, 0 < β → ∀ᶠ ε : ℝ in 𝓝[>] 0,
SweepsCover ε (RadialCaps.linearCap β ε))
(hcubic : ∀ c : ℝ, 9 / 2 ≤ c → ∀ ε : ℝ, 0 < ε →
SweepsCover ε (RadialCaps.cubicCap c ε))
(hfractional : ∀ c : ℝ, 0 < c → ∀ᶠ ε : ℝ in 𝓝[>] 0,
SweepsCover ε (RadialCaps.fractionalCap c ε)) : LCFStatement := by
refine ⟨?_, ?_, ?_⟩
· intro β hβ
have hcov : ∀ᶠ ε : ℝ in 𝓝[>] 0,
SweepsCover ε (RadialCaps.linearCap β ε) ∧ CoverLimit ε (RadialCaps.linearCap β ε) := by
filter_upwards [hlinear β hβ] with ε hε
exact ⟨hε, coverLimit_of_paired hpaired (RadialCaps.continuous_linearCap β ε).continuousOn
(fun r _ => (RadialCaps.linearCap_pos hβ.le ε r).le) hε⟩
refine ⟨hcov, RadialCaps.linear_cost_bigO hβ.le, ?_⟩
intro hβ1
filter_upwards [hcov, RadialCaps.linear_cost_strict hβ hβ1] with ε hε hQ
exact ⟨hQ, finiteSaving_of_coverLimit hε.2 hQ⟩
· intro c hc
have hc0 : 0 ≤ c := by linarith
have hcov : ∀ ε : ℝ, 0 < ε →
SweepsCover ε (RadialCaps.cubicCap c ε) ∧ CoverLimit ε (RadialCaps.cubicCap c ε) := by
intro ε hε
have h := hcubic c hc ε hε
exact ⟨h, coverLimit_of_paired hpaired (RadialCaps.continuous_cubicCap c ε).continuousOn
(fun r _ => (RadialCaps.cubicCap_pos hc0 hε.le r).le) h⟩
refine ⟨hcov, RadialCaps.cubic_cost_bigO hc0, ?_⟩
filter_upwards [RadialCaps.eventually_small_scale c, RadialCaps.cubic_cost_strict hc0] with ε hε hQ
exact ⟨hQ, finiteSaving_of_coverLimit (hcov ε hε.1).2 hQ⟩
· intro c hc
have hcov : ∀ᶠ ε : ℝ in 𝓝[>] 0,
SweepsCover ε (RadialCaps.fractionalCap c ε) ∧ CoverLimit ε (RadialCaps.fractionalCap c ε) := by
filter_upwards [hfractional c hc, RadialCaps.eventually_small_scale c] with ε hε hs
exact ⟨hε, coverLimit_of_paired hpaired (RadialCaps.continuous_fractionalCap c ε).continuousOn
(fun r _ => (RadialCaps.fractionalCap_pos hc.le hs.1.le r).le) hε⟩
refine ⟨hcov, RadialCaps.fractional_cost_bigO hc.le, ?_⟩
filter_upwards [hcov, RadialCaps.fractional_cost_strict hc.le] with ε hε hQ
exact ⟨hQ, finiteSaving_of_coverLimit hε.2 hQ⟩
lemma eventually_positive_le {a : ℝ} (ha : 0 < a) :
∀ᶠ ε : ℝ in 𝓝[>] 0, 0 < ε ∧ ε ≤ a := by
have hlt : ∀ᶠ ε : ℝ in 𝓝 0, ε < a :=
continuousAt_id.eventually_lt continuousAt_const ha
filter_upwards [eventually_mem_nhdsWithin,
Filter.Eventually.filter_mono nhdsWithin_le_nhds hlt] with ε hε hεa
exact ⟨hε, hεa.le⟩
theorem radial_cap_variants : LCFStatement := by
apply assemble RadialPairedSweep.paired_radial_sweeps
· intro β hβ
filter_upwards [eventually_positive_le (show 0 < 2 * β / 9 by positivity)] with ε hε
exact RadialCapCoverage.linear_covers hβ.le hε.1 hε.2
· intro c hc ε hε
exact RadialCapCoverage.cubic_covers hc hε
· intro c hc
filter_upwards [eventually_positive_le
(show 0 < (2 * c / 9) ^ 2 by positivity)] with ε hε
exact RadialCapCoverage.fractional_covers hc hε.1 hε.2
end CylinderCovering.RadialCapVariants
end
end OAI
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