File size: 2,636 Bytes
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 | import OAI.Geometry.CylinderCovering.Angular.CubicScaledFamily
import OAI.Geometry.CylinderCovering.Angular.CubicCost
import OAI.Geometry.CylinderCovering.Cubic.Similarity
namespace OAI
noncomputable section
namespace CylinderCovering.Cubic
open Set TriangularCovering
theorem main_proof : MainStatement := by
have hmin : minimumProjection4 = 4 * H := K4_projection_minimum
refine ⟨hmin, 3208 * H, 1 / 100000, mul_pos (by norm_num) H_pos, by norm_num, ?_⟩
intro ε hε hsmall
obtain ⟨c, hcover, harea, hpres⟩ := finiteCover_proof ε hε hsmall
refine ⟨c, hcover, hpres, harea, ?_, ?_⟩
· rw [harea]
have h := cubicCost_expansion hε hsmall
have hid : cubicCost ε - (2 * H - H / 15 * ε ^ 2) =
H * (cubicCost ε / H - (2 - ε ^ 2 / 15)) := by
field_simp [H_ne_zero]
rw [hid, abs_mul, abs_of_pos H_pos]
nlinarith only [mul_le_mul_of_nonneg_left h H_pos.le]
· intro hεsmall
rw [harea, hmin]
have h := cubicCost_strict hε hεsmall.le
linarith
theorem scaledMain_proof : ScaledMainStatement := by
refine ⟨401 / 4, 1 / 50000, by norm_num, by norm_num, ?_⟩
intro τ hτ hsmall
obtain ⟨c, hcover, harea, hpres⟩ := scaledFiniteCover_proof τ hτ hsmall
refine ⟨c, hcover, hpres, harea, ?_, ?_⟩
· rw [harea]
have hid : cubicCost (τ / 2) / 4 / H = cubicCost (τ / 2) / (4 * H) := by ring
rw [hid]
exact scaled_cubicCost_expansion hτ hsmall
· intro hτsmall
rw [harea, TriangularCovering.main.1]
have h := cubicCost_strict (show 0 < τ / 2 by linarith)
(show τ / 2 ≤ 1 / 100000 by linarith)
linarith
lemma relativeCost4_eq {m : ℕ} (c : Fin m → TriCylinder) :
relativeCost4 c = relativeCost c / 4 := by
unfold relativeCost4 relativeCost
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro i _
rw [K4_shadow]
ring
lemma relativeCost4_lt_of_total_lt {m : ℕ} (c : Fin m → TriCylinder)
(h : totalArea c < minimumProjection4 / 2) : relativeCost4 c < 1 / 2 := by
have hmin : minimumProjection4 = 4 * H := K4_projection_minimum
rw [hmin] at h
have htotal : totalArea c / H < 2 := (div_lt_iff₀ H_pos).mpr (by linarith)
have hrelative := relativeCost_le_totalArea c
rw [relativeCost4_eq]
linarith
lemma finiteFamily4_normalized {ε : ℝ} (hε : 0 < ε) (hsmall : ε ≤ 1 / 100000) :
relativeCost4 (finiteFamily4 ε hε) < 1 / 2 := by
apply relativeCost4_lt_of_total_lt
rw [finiteFamily4_total]
have hmin : minimumProjection4 = 4 * H := K4_projection_minimum
rw [hmin]
have h := cubicCost_strict hε hsmall
linarith
end CylinderCovering.Cubic
end
end OAI
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