| import OAI.Geometry.CylinderCovering.Cartesian.FullDefinitions |
| import OAI.Geometry.CylinderCovering.Cartesian.Statement |
| import OAI.Geometry.CylinderCovering.Cartesian.Cover |
|
|
| namespace OAI |
|
|
| open Set |
| namespace CylinderCovering.CartesianCaps |
|
|
| theorem cutoff_essential : CartesianCutoffStatement := by |
| intro τ η q hτ hη hq hq1 |
| have hη0 : 0 < η := hη.1 |
| obtain ⟨B, hB, htail⟩ := untruncated_contains_ray τ η q hτ.1 hη.1.le hq |
| have hzero : (0 : ℝ) ∈ untruncated τ η q := by |
| constructor |
| · norm_num |
| · simp only [mul_zero, height, D, zero_pow (by omega : 2 ≠ 0), |
| zero_pow (by omega : 4 ≠ 0), zero_div, zero_add] |
| positivity |
| have hfar : B ∈ untruncated τ η q := |
| ⟨by linarith, htail B le_rfl⟩ |
| refine ⟨untruncated_not_bddAbove τ η q hτ.1 hη.1.le hq, ?_, B, hB, ?_⟩ |
| · intro hc |
| have hone := hc.Icc_subset hzero hfar (show (1 : ℝ) ∈ Icc 0 B from ⟨by norm_num, hB⟩) |
| exact untruncated_gap τ η q ⟨hτ.1.le,hτ.2⟩ hη.2 hq1 hone.2 |
| · intro s hs |
| change B ≤ s at hs |
| exact ⟨by linarith, htail s hs⟩ |
|
|
| theorem cartesian_full : CartesianFullStatement := |
| ⟨cartesianAperture_proved, cartesian_cover, cutoff_essential⟩ |
|
|
| end CylinderCovering.CartesianCaps |
|
|
| end OAI |
|
|
