| import OAI.Geometry.CylinderCovering.Beams.BeamStrip |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped InnerProductSpace |
| namespace CylinderCovering.BeamStrip |
|
|
| theorem rectangle_isRectangle {q A l r : ℝ} (hA : 0 < A) (hlr : l < r) : |
| IsRectangle (rectangle q A l r) := by |
| refine ⟨(-A) • normal q + l • vertical, (2*A) • normal q, |
| (r-l) • vertical, ?_, ?_, ?_, ?_⟩ |
| · simp [inner_smul_left, inner_smul_right, normal_vertical_inner] |
| · exact smul_ne_zero (by positivity) (by |
| intro h; have := normal_norm q; rw [h, norm_zero] at this; norm_num at this) |
| · exact smul_ne_zero (sub_ne_zero.mpr hlr.ne') (by |
| intro h; have := vertical_norm; rw [h, norm_zero] at this; norm_num at this) |
| · ext z |
| constructor |
| · rintro ⟨s, hs, t, ht, rfl⟩ |
| refine ⟨(s+A)/(2*A), ?_, (t-l)/(r-l), ?_, ?_⟩ |
| · constructor |
| · exact div_nonneg (by linarith [hs.1]) (by positivity) |
| · apply (div_le_one (by positivity : 0 < 2*A)).mpr; linarith [hs.2] |
| · constructor |
| · exact div_nonneg (by linarith [ht.1]) (by linarith) |
| · apply (div_le_one (by linarith : 0 < r-l)).mpr; linarith [ht.2] |
| · have h1 : (s+A)/(2*A)*(2*A) = s+A := div_mul_cancel₀ _ (by positivity) |
| have h2 : (t-l)/(r-l)*(r-l) = t-l := div_mul_cancel₀ _ (sub_ne_zero.mpr hlr.ne') |
| simp only [smul_smul, h1, h2] |
| module |
| · rintro ⟨s, hs, t, ht, rfl⟩ |
| refine ⟨-A+s*(2*A), ?_, l+t*(r-l), ?_, ?_⟩ |
| · constructor <;> nlinarith [mul_nonneg hs.1 hA.le, |
| mul_nonneg (sub_nonneg.mpr hs.2) hA.le] |
| · constructor <;> nlinarith [mul_nonneg ht.1 (sub_nonneg.mpr hlr.le), |
| mul_nonneg (sub_nonneg.mpr ht.2) (sub_nonneg.mpr hlr.le)] |
| · simp only [smul_smul] |
| module |
|
|
| theorem IsHorizontalRectangle.hasRectangularBase {C : RuledApproximation.Cylinder} |
| (hC : IsHorizontalRectangle C) : HasRectangularBase C := by |
| obtain ⟨q,A,l,r,hA,hlr,_,hbase⟩ := hC |
| unfold HasRectangularBase |
| rw [hbase] |
| exact rectangle_isRectangle hA hlr |
|
|
| end CylinderCovering.BeamStrip |
|
|
| end |
|
|
| end OAI |
|
|
