| import OAI.Geometry.CylinderCovering.Cartesian.CoverEligibility | |
| import OAI.Geometry.CylinderCovering.Angular.GeneralTheorem | |
| namespace OAI | |
| open Set | |
| namespace CylinderCovering.CartesianCaps | |
| theorem cartesian_cover : CartesianCoverStatement := by | |
| intro η hη hη480 M hM | |
| obtain ⟨δ, A, hδ, hA, hg⟩ := | |
| AngularGeneral.general_cap η hη hη480 (max M 12) (le_max_of_le_right (by norm_num)) | |
| refine ⟨min δ (1/4), A, lt_min hδ (by norm_num), hA, ?_⟩ | |
| intro τ hτ hτδ N a hN ha0 haN ha hmesh | |
| have ht : τ ∈ Icc (0 : ℝ) (1/4) := | |
| ⟨hτ.le, (hτδ.trans_le (min_le_right _ _)).le⟩ | |
| have he : η ∈ Ioc (0 : ℝ) 1 := ⟨hη, by linarith⟩ | |
| have hd := cartesianData_admissible hN ha0 haN ha ht he hM hmesh | |
| refine ⟨hg τ hτ (hτδ.trans_le (min_le_left _ _)) _ hd, ?_⟩ | |
| intro b hb j hj | |
| have hraw := AngularOrdinaryCaps.ordinaryData_admissible .radial | |
| hN ha0 haN ha hη.le hM hmesh | |
| have hn := hraw.2.2.2.2.1 | |
| constructor | |
| · exact cartesian_lowerIntercept τ η b | |
| (AngularOrdinaryCaps.ordinaryData .radial N a τ η) ht he hb hn j hj | |
| · exact cartesian_upperIntercept τ η b | |
| (AngularOrdinaryCaps.ordinaryData .radial N a τ η) ht he hb hn j hj | |
| end CylinderCovering.CartesianCaps | |
| end OAI | |