| import OAI.Geometry.CylinderCovering.Radial.OverlapQuantitativeDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.OverlapBounds |
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| namespace OAI |
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| noncomputable section |
| namespace CylinderCovering.RadialOverlapQuantitative |
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| open RadialOverlap RadialOverlapBounds |
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| theorem canonical : CanonicalStatement := by |
| intro eps x y t p q heps hx hy hp hq hroots hrho hsigma |
| obtain ⟨hrootq, hrootp⟩ := hroots |
| have hl : eps * y * p ≤ t := by |
| dsimp [sigma] at hsigma |
| linarith only [hsigma] |
| have hu : t ≤ eps * x * q := by |
| dsimp [rho] at hrho |
| linarith only [hrho] |
| obtain ⟨ht, hxp, hyq⟩ := root_estimates heps.le hx hy hp hq hrootq hrootp hl hu |
| obtain ⟨_, hsum, hr0, hr, hs0, hs, hdiff⟩ := |
| joint_radial_bounds heps.le hx hy hp hq hrootq hrootp hrho hsigma |
| exact ⟨ht, hxp, hyq, excess_identity eps x y t p q, hr0, hr, hs0, hs, |
| hsum, hdiff.trans hsum, interface_error heps.le hx hy hp hq hrootq hrootp hl hu, |
| cap_comparison heps.le hx hy hp hq hrootq hrootp hl hu, |
| cubic_twelve_compensation heps hx hy hp hq hrootq hrootp hl hu⟩ |
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| theorem physical : PhysicalStatement := by |
| intro tau x y z p q htau hx hy hp hq hrootq hrootp hl hu |
| have hx2 : |x| ≤ 2 := by linarith only [hx, hy, abs_nonneg y] |
| have hy2 : |y| ≤ 2 := by linarith only [hx, hy, abs_nonneg x] |
| exact ⟨physical_cap_comparison (by positivity) htau.le hx2 hy2 hp hq hrootq hrootp hl hu, |
| orthonormal_height_compensation htau hx2 hy2 hp hq hrootq hrootp hl hu⟩ |
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| theorem quantitative_overlap : Statement := ⟨canonical, physical⟩ |
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| end CylinderCovering.RadialOverlapQuantitative |
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| end |
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| end OAI |
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