| import OAI.Geometry.CylinderCovering.Rational.RealizationDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.RationalThreshold |
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| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped Topology |
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| namespace SlopeRational |
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| 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 |
|
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| 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 |
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| end SlopeRational |
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| end |
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| end OAI |
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