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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 34 35 36 | 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
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