| import OAI.Geometry.CylinderCovering.Angular.Margins |
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|
| namespace OAI |
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|
| noncomputable section |
| namespace CylinderCovering.Cubic |
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|
| theorem profileMax_eq_sSup (hab : a ≤ b) : |
| profileMax a b = sSup (profile '' Set.Icc a b) := by |
| have hbdd : BddAbove (profile '' Set.Icc a b) := by |
| refine ⟨profileMax a b, ?_⟩ |
| rintro z ⟨q, hq, rfl⟩ |
| exact profile_le_max_endpoints hq.1 hq.2 |
| apply le_antisymm |
| · apply max_le |
| · exact le_csSup hbdd ⟨a, ⟨le_rfl, hab⟩, rfl⟩ |
| · exact le_csSup hbdd ⟨b, ⟨hab, le_rfl⟩, rfl⟩ |
| · apply csSup_le |
| · exact ⟨profile a, a, ⟨le_rfl, hab⟩, rfl⟩ |
| · rintro z ⟨q, hq, rfl⟩ |
| exact profile_le_max_endpoints hq.1 hq.2 |
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|
| theorem profileMax_attained (hab : a ≤ b) : |
| ∃ q ∈ Set.Icc a b, profile q = profileMax a b := by |
| rcases le_total (profile a) (profile b) with h | h |
| · exact ⟨b, ⟨hab, le_rfl⟩, (max_eq_right h).symm⟩ |
| · exact ⟨a, ⟨le_rfl, hab⟩, (max_eq_left h).symm⟩ |
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|
| theorem profileMax_isGreatest (hab : a ≤ b) : |
| IsGreatest (profile '' Set.Icc a b) (profileMax a b) := by |
| constructor |
| · obtain ⟨q, hq, heq⟩ := profileMax_attained hab |
| exact ⟨q, hq, heq⟩ |
| · rintro z ⟨q, hq, rfl⟩ |
| exact profile_le_max_endpoints hq.1 hq.2 |
|
|
| end CylinderCovering.Cubic |
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|
| end |
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| end OAI |
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|
