| import OAI.Geometry.CylinderCovering.Beams.MainDefinitions | |
| namespace OAI | |
| noncomputable section | |
| open Set | |
| open scoped InnerProductSpace | |
| namespace CylinderCovering.BeamStrip | |
| theorem IsRectangle.isCompact {B : Set Space} (hB : IsRectangle B) : IsCompact B := by | |
| obtain ⟨c,u,v,_,_,_,rfl⟩ := hB | |
| have heq : {z | ∃ s ∈ Icc (0:ℝ) 1, ∃ t ∈ Icc (0:ℝ) 1, | |
| z = c+s • u+t • v} = | |
| (fun p : ℝ × ℝ => c+p.1 • u+p.2 • v) '' (Icc (0:ℝ) 1 ×ˢ Icc (0:ℝ) 1) := by | |
| ext z | |
| constructor | |
| · rintro ⟨s,hs,t,ht,rfl⟩ | |
| exact ⟨(s,t),⟨hs,ht⟩,rfl⟩ | |
| · rintro ⟨⟨s,t⟩,⟨hs,ht⟩,rfl⟩ | |
| exact ⟨s,hs,t,ht,rfl⟩ | |
| rw [heq] | |
| exact (isCompact_Icc.prod isCompact_Icc).image (by fun_prop) | |
| theorem HasRectangularBase.isCompact {C : RuledApproximation.Cylinder} | |
| (hC : HasRectangularBase C) : IsCompact C.base := | |
| Subtype.isCompact_iff.mpr (IsRectangle.isCompact hC) | |
| end CylinderCovering.BeamStrip | |
| namespace ShortenedBeam | |
| theorem HasTriangularBase.isCompact {C : RuledApproximation.Cylinder} | |
| (hC : HasTriangularBase C) : IsCompact C.base := by | |
| obtain ⟨v,hv,hbase⟩ := hC | |
| rw [Subtype.isCompact_iff,hbase] | |
| exact (Set.finite_range v).isCompact_convexHull ℝ | |
| theorem HasMixedBase.isCompact {C : RuledApproximation.Cylinder} | |
| (hC : HasMixedBase C) : IsCompact C.base := by | |
| rcases hC with hC | hC | |
| · exact hC.isCompact | |
| · exact hC.isCompact | |
| end ShortenedBeam | |
| end | |
| end OAI | |