| import OAI.Geometry.CylinderCovering.Radial.SqrtRealizationDefinitions | |
| import OAI.Geometry.CylinderCovering.Radial.SqrtCap | |
| import OAI.Geometry.CylinderCovering.Radial.FinitePair | |
| import OAI.Geometry.CylinderCovering.Radial.PairedSweep | |
| import OAI.Geometry.CylinderCovering.Radial.CapCoverage | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering.RadialSqrtCap | |
| open Set Filter | |
| open scoped Topology | |
| theorem exists_coverage_threshold {β : ℝ} (hβ : 0 < β) : | |
| ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, 0 < ε → ε < δ → | |
| RadialPairedSweep.tetrahedron (Real.sqrt 2) ⊆ | |
| RadialPairedSweep.upperSweep (Real.sqrt 2) ε (cap β ε) ∪ | |
| RadialPairedSweep.lowerSweep (Real.sqrt 2) ε (cap β ε) := by | |
| obtain ⟨δ, hδ, hpad⟩ := exists_padding_threshold hβ | |
| refine ⟨δ, hδ, ?_⟩ | |
| intro ε he hεδ | |
| obtain ⟨heβ, hcap⟩ := hpad ε he hεδ | |
| have hlinear := RadialCapCoverage.linear_covers (h := Real.sqrt 2) | |
| (beta := β / 2) (by positivity) he (by linarith only [heβ]) | |
| have hmono : ∀ q ∈ Icc (-1 : ℝ) 1, | |
| RadialCaps.linearCap (β / 2) ε q ≤ cap β ε q := by | |
| intro q hq | |
| exact hcap q (abs_le.mpr hq) | |
| exact hlinear.trans (Set.union_subset_union | |
| (RadialPairedSweep.upperSweep_mono _ _ hmono) | |
| (RadialPairedSweep.lowerSweep_mono _ _ hmono)) | |
| theorem finite_family {β ε : ℝ} | |
| (hcover : RadialPairedSweep.tetrahedron (Real.sqrt 2) ⊆ | |
| RadialPairedSweep.upperSweep (Real.sqrt 2) ε (cap β ε) ∪ | |
| RadialPairedSweep.lowerSweep (Real.sqrt 2) ε (cap β ε)) | |
| (tagU tagL : (N : ℕ) → Fin N → ℝ) | |
| (htagU : ∀ N, RadialRiemann.IsTagged (-1) 1 N (tagU N)) | |
| (htagL : ∀ N, RadialRiemann.IsTagged (-1) 1 N (tagL N)) : | |
| ∃ C : (N : ℕ) → Fin (2 * N) → TriangularCovering.TriCylinder, | |
| FamilyConclusion β ε tagU tagL C := by | |
| obtain ⟨U, L, hUL, hlim⟩ := RadialPairedSweep.paired_radial_sweeps | |
| (Real.sqrt 2) ε (by positivity) (cap β ε) | |
| (cap_continuous β ε).continuousOn (fun q _ => cap_nonneg β ε q) | |
| tagU tagL htagU htagL | |
| refine ⟨fun N => RadialPairedSweep.join (U N) (L N), ?_, ?_⟩ | |
| · intro N hN | |
| obtain ⟨haxes, hU, hL⟩ := hUL N hN | |
| have hwhole := RadialPairedSweep.join_covers_union (U N) (L N) hU hL | |
| refine ⟨?_, ?_, hwhole⟩ | |
| · intro i | |
| simpa only [RadialPairedSweep.join_upper, RadialPairedSweep.join_lower] using haxes i | |
| · intro z hz | |
| exact hwhole z (hcover hz) | |
| · rw [RadialPairedSweep.physicalCost_sqrt_two] at hlim | |
| exact RadialPairedSweep.join_totalArea_tendsto U L hlim | |
| theorem full_square_root_cap : FullStatement := by | |
| refine ⟨fun β ε hβ he => cost_expansion hβ he, ?_, ?_⟩ | |
| · intro β hβ | |
| obtain ⟨δ, hδ, hcover⟩ := exists_coverage_threshold hβ | |
| refine ⟨δ, hδ, ?_⟩ | |
| intro ε he hεδ | |
| have hc := hcover ε he hεδ | |
| exact ⟨hc, fun tagU tagL htagU htagL => finite_family hc tagU tagL htagU htagL⟩ | |
| · intro β hβ hβmax | |
| obtain ⟨δc, hδc, hcover⟩ := exists_coverage_threshold hβ | |
| obtain ⟨δs, hδs, hsave⟩ := exists_saving_threshold hβ hβmax | |
| refine ⟨min δc δs, lt_min hδc hδs, ?_⟩ | |
| intro ε he hεδ | |
| have hc := hcover ε he (lt_of_lt_of_le hεδ (min_le_left _ _)) | |
| have hs := hsave ε he (lt_of_lt_of_le hεδ (min_le_right _ _)) | |
| refine ⟨hs, ?_⟩ | |
| intro tagU tagL htagU htagL | |
| obtain ⟨C, hC⟩ := finite_family hc tagU tagL htagU htagL | |
| refine ⟨C, hC, ?_⟩ | |
| apply RadialPairedSweep.eventually_strict_finite_cost C hC.2 | |
| have hp : 0 < 2 * Real.sqrt 2 := by positivity | |
| simpa only [mul_one] using mul_lt_mul_of_pos_left hs hp | |
| end CylinderCovering.RadialSqrtCap | |
| end | |
| end OAI | |