| import OAI.Geometry.CylinderCovering.Physical.Paired |
| import OAI.Geometry.CylinderCovering.Radial.SqrtCap |
| import OAI.Geometry.CylinderCovering.Radial.CapCoverage |
|
|
| namespace OAI |
|
|
| open Set Filter |
| open scoped Topology |
|
|
| noncomputable section |
| namespace CylinderCovering.UnitEdgeFootprints |
| open TriangularCovering |
|
|
| lemma cap_dictionary (η q : ℝ) : |
| RadialSqrtCap.cap (1 / 120) (η / 2) q = 2 * sourceCap η (q / 2) := by |
| have hleft : 0 ≤ RadialSqrtCap.cap (1 / 120) (η / 2) q := Real.sqrt_nonneg _ |
| have hright : 0 ≤ 2 * sourceCap η (q / 2) := mul_nonneg (by norm_num) (sourceCap_pos η (q / 2)).le |
| apply (sq_eq_sq₀ hleft hright).mp |
| rw [RadialSqrtCap.cap_sq (by norm_num : (0 : ℝ) ≤ 1 / 120), mul_pow] |
| have hs : (sourceCap η (q / 2)) ^ 2 = |
| 1 / 4 + η ^ 2 * ((q / 2) ^ 2 / 4 + (q / 2) ^ 4 + 1 / 960) := by |
| unfold sourceCap |
| exact Real.sq_sqrt (by positivity) |
| rw [hs] |
| unfold RadialSqrtCap.profile RadialCaps.H |
| ring |
|
|
| theorem source_coverage : CoverageStatement := by |
| obtain ⟨δ, hδ, hpad⟩ := RadialSqrtCap.exists_padding_threshold |
| (by norm_num : (0 : ℝ) < 1 / 120) |
| refine ⟨2 * δ, by positivity, ?_⟩ |
| intro η hη hsmall z hz |
| obtain ⟨t, ht, hz2, hx, hy⟩ := hz |
| have he : 0 < η / 2 := by positivity |
| obtain ⟨hεβ, hcap⟩ := hpad (η / 2) he (by linarith) |
| have hx' : |4 * z 1| ≤ 1 + (2 * t - 1) := by |
| rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 4)] |
| linarith |
| have hy' : |4 * z 0| ≤ 1 - (2 * t - 1) := by |
| rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 4)] |
| linarith |
| have hc := RadialCapCoverage.padding_coverage (eps := η / 2) |
| (R := RadialCaps.linearCap (1 / 240) (η / 2)) (b := fun _ => (1 / 240 : ℝ)) he |
| (fun _ _ => rfl) (fun _ _ => by norm_num) |
| (fun _ _ _ _ => by linarith) hx' hy' |
| rcases hc with hu | hl |
| · obtain ⟨q, ρ, hq, hρ, hR, hrep, htrep⟩ := hu |
| right |
| have hqI := abs_le.mp hq |
| have hcapq := hcap q hq |
| have hρcap : ρ ≤ 2 * sourceCap η (q / 2) := by |
| rw [← cap_dictionary] |
| apply hR.trans |
| simpa only [RadialCaps.linearCap, show (1 / 120 : ℝ) / 2 = 1 / 240 by norm_num] using hcapq |
| refine ⟨q / 2, ⟨by linarith, by linarith⟩, ρ / 2, |
| ⟨by positivity, by linarith⟩, z 1, ?_, ?_⟩ |
| · linarith [ht.2] |
| · ext i |
| fin_cases i |
| · change z 0 = q / 2 * (ρ / 2) - η * z 1 * (1 / 4 + (q / 2) ^ 2) |
| unfold RadialCaps.G at hrep |
| nlinarith only [hrep] |
| · rfl |
| · change z 2 = height * (1 - ρ / 2 + 2 * η * z 1 * (q / 2)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [htrep] |
| · obtain ⟨q, ρ, hq, hρ, hR, hrep, htrep⟩ := hl |
| left |
| have hqI := abs_le.mp hq |
| have hcapq := hcap q hq |
| have hρcap : ρ ≤ 2 * sourceCap η (q / 2) := by |
| rw [← cap_dictionary] |
| apply hR.trans |
| simpa only [RadialCaps.linearCap, show (1 / 120 : ℝ) / 2 = 1 / 240 by norm_num] using hcapq |
| refine ⟨q / 2, ⟨by linarith, by linarith⟩, ρ / 2, |
| ⟨by positivity, by linarith⟩, z 0, ?_, ?_⟩ |
| · linarith [ht.1] |
| · ext i |
| fin_cases i |
| · rfl |
| · change z 1 = q / 2 * (ρ / 2) + η * z 0 * (1 / 4 + (q / 2) ^ 2) |
| unfold RadialCaps.G at hrep |
| nlinarith only [hrep] |
| · change z 2 = height * (ρ / 2 + 2 * η * z 0 * (q / 2)) |
| rw [hz2] |
| congr 1 |
| nlinarith only [htrep] |
|
|
| theorem finite_strict_of_coverage (hcoverage : CoverageStatement) : FiniteStrictStatement := by |
| obtain ⟨δ, hδ, hcover⟩ := hcoverage |
| refine ⟨min δ (1 / 40), by positivity, ?_⟩ |
| intro η hη hsmall |
| have hc := hcover η hη (lt_of_lt_of_le hsmall (min_le_left _ _)) |
| have hs := source_cost_strict hη (le_trans hsmall.le (min_le_right _ _)) |
| obtain ⟨C₁, C₂, hspec, hpair, hlim⟩ := paired_family_realization η hη |
| have hevent : ∀ᶠ N in atTop, totalArea (C₁ N) + totalArea (C₂ N) < height / 4 := |
| hlim.eventually (gt_mem_nhds hs) |
| obtain ⟨N, hcost⟩ := (eventually_atTop.1 hevent) |
| let M := max N 1 |
| have hM : 0 < M := lt_of_lt_of_le (by norm_num : 0 < 1) (le_max_right _ _) |
| refine ⟨M, hM, C₁ M, C₂ M, ?_, ?_, hcost M (le_max_left _ _)⟩ |
| · intro i |
| obtain ⟨h1, h2, h3, h4, _, _⟩ := hspec M i |
| exact ⟨h1, h2, h3, h4⟩ |
| · intro z hz |
| exact hpair M hM z (hc hz) |
|
|
| lemma tetrahedron_similarity : tetrahedron = |
| Similarity.map (0 : Space) (1 / 2) (LinearIsometryEquiv.refl ℝ Space) '' K := by |
| have hh : H / 2 = height := by |
| unfold H height |
| have hs : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num) |
| have hn : Real.sqrt 2 ≠ 0 := by positivity |
| field_simp |
| nlinarith only [hs] |
| ext z |
| constructor |
| · rintro ⟨t, ht, hz, hx, hy⟩ |
| refine ⟨point (2 * z 0) (2 * z 1) t, ?_, ?_⟩ |
| · refine ⟨2 * z 0, 2 * z 1, t, ht.1, ht.2, ?_, ?_, rfl⟩ |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)]; linarith |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)]; linarith |
| · ext i |
| fin_cases i |
| · simp [Similarity.map, point] |
| · simp [Similarity.map, point] |
| · change 0 + (1 / 2 : ℝ) * (H * t) = z 2 |
| rw [hz, ← hh] |
| ring |
| · rintro ⟨_, ⟨x, y, t, ht0, ht1, hx, hy, rfl⟩, rfl⟩ |
| refine ⟨t, ⟨ht0, ht1⟩, ?_, ?_, ?_⟩ |
| · change 0 + (1 / 2 : ℝ) * (H * t) = height * t |
| rw [← hh] |
| ring |
| · change |0 + (1 / 2 : ℝ) * x| ≤ (1 - t) / 2 |
| rw [zero_add, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)] |
| linarith |
| · change |0 + (1 / 2 : ℝ) * y| ≤ t / 2 |
| rw [zero_add, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)] |
| linarith |
|
|
| theorem minimum_area : Similarity.minimumArea tetrahedron = height / 2 := by |
| rw [tetrahedron_similarity, Similarity.minimumArea_map _ _ |
| (by norm_num : (1 / 2 : ℝ) ≠ 0) (LinearIsometryEquiv.refl ℝ Space)] |
| change (1 / 2 : ℝ) ^ 2 * A_min = height / 2 |
| rw [projection_minimum] |
| unfold H height |
| have hs : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num) |
| have hn : Real.sqrt 2 ≠ 0 := by positivity |
| field_simp |
| nlinarith only [hs] |
|
|
| theorem full_unit_edge : FullStatement := |
| ⟨local_realization, family_realization, paired_family_realization, source_cost, |
| source_coverage, finite_strict_of_coverage source_coverage, minimum_area⟩ |
|
|
| end CylinderCovering.UnitEdgeFootprints |
|
|
| end |
|
|
| end OAI |
|
|
