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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 | 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
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