| import OAI.Geometry.CylinderCovering.Cartesian.GeneralDefinitions |
| import OAI.Geometry.CylinderCovering.Cartesian.Caps |
|
|
| namespace OAI |
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|
| open Set |
| namespace CylinderCovering.CartesianCaps |
|
|
| lemma interval_subset_unit {D : AngularGeneral.Data} |
| (hnodes : ∀ j ≤ D.count, |D.node j| ≤ 1) {j : ℕ} (hj : j < D.count) : |
| AngularGeneral.interval D j ⊆ Icc (-1 : ℝ) 1 := by |
| intro q hq |
| have hl := abs_le.mp (hnodes j hj.le) |
| have hr := abs_le.mp (hnodes (j+1) (by omega)) |
| exact ⟨hl.1.trans hq.1, hq.2.trans hr.2⟩ |
|
|
| theorem cartesian_admissible (τ η E : ℝ) (D : AngularGeneral.Data) |
| (hτ : τ ∈ Icc (0 : ℝ) (1/4)) (hη : η ∈ Ioc (0 : ℝ) 1) |
| (hD : AngularGeneral.Admissible τ η E D) : |
| AngularGeneral.Admissible τ η (max E 12) (withCartesianCaps τ η D) := by |
| rcases hD with ⟨hn, hzero, hlast, hstrict, hnodes, hboundary, hcontinuous, hpositive, herr⟩ |
| refine ⟨hn, hzero, hlast, hstrict, hnodes, hboundary, ?_, ?_, ?_⟩ |
| · intro j hj |
| exact (cap_continuousOn τ η hτ hη).mono (interval_subset_unit hnodes hj) |
| · intro j hj q hq |
| have haq : |q| ≤ 1 := abs_le.mpr (interval_subset_unit hnodes hj hq) |
| have h := (cap_spec τ η q hτ hη haq).1.1 |
| change 0 < cap τ η q |
| linarith |
| · intro j hj q hq |
| have hqD : q ∈ AngularGeneral.interval D j := hq |
| have he := herr j hj q hqD |
| have hE := mul_le_mul_of_nonneg_right (le_max_left E 12) (sq_nonneg τ) |
| have h12 := mul_le_mul_of_nonneg_right (le_max_right E 12) (show 0 ≤ τ^4 by positivity) |
| have haq : |q| ≤ 1 := abs_le.mpr (interval_subset_unit hnodes hj hqD) |
| refine ⟨he.1.trans hE, he.2.1.trans hE, ?_⟩ |
| exact (cap_fourth_order τ η q hτ hη haq).trans h12 |
|
|
| lemma cartesian_lowerIntercept (τ η b : ℝ) (D : AngularGeneral.Data) |
| (hτ : τ ∈ Icc (0 : ℝ) (1/4)) (hη : η ∈ Ioc (0 : ℝ) 1) |
| (hb : b ∈ Icc (3/4 : ℝ) 1) (hnodes : ∀ j ≤ D.count, |D.node j| ≤ 1) |
| (j : ℕ) (hj : j < D.count) : |
| AngularGeneral.lowerIntercept (withCartesianCaps τ η D) j = |
| {z | ∃ q ∈ AngularGeneral.interval D j, ∃ s ∈ aperture τ η b q, |
| z = TriangularCovering.point 0 (q*s) s} := by |
| ext z |
| change (∃ q ∈ AngularGeneral.interval D j, ∃ s ∈ Icc 0 (cap τ η q), |
| z = TriangularCovering.point 0 (q*s) s) ↔ _ |
| constructor |
| · rintro ⟨q,hq,s,hs,hz⟩ |
| refine ⟨q,hq,s,?_,hz⟩ |
| rwa [aperture_eq_Icc τ η b q hτ hη hb (abs_le.mpr (interval_subset_unit hnodes hj hq))] |
| · rintro ⟨q,hq,s,hs,hz⟩ |
| refine ⟨q,hq,s,?_,hz⟩ |
| rw [aperture_eq_Icc τ η b q hτ hη hb (abs_le.mpr (interval_subset_unit hnodes hj hq))] at hs |
| exact hs |
|
|
| lemma cartesian_upperIntercept (τ η b : ℝ) (D : AngularGeneral.Data) |
| (hτ : τ ∈ Icc (0 : ℝ) (1/4)) (hη : η ∈ Ioc (0 : ℝ) 1) |
| (hb : b ∈ Icc (3/4 : ℝ) 1) (hnodes : ∀ j ≤ D.count, |D.node j| ≤ 1) |
| (j : ℕ) (hj : j < D.count) : |
| AngularGeneral.upperIntercept (withCartesianCaps τ η D) j = |
| {z | ∃ q ∈ AngularGeneral.interval D j, ∃ s ∈ aperture τ η b q, |
| z = TriangularCovering.point (q*s) 0 (1-s)} := by |
| ext z |
| change (∃ q ∈ AngularGeneral.interval D j, ∃ s ∈ Icc 0 (cap τ η q), |
| z = TriangularCovering.point (q*s) 0 (1-s)) ↔ _ |
| constructor |
| · rintro ⟨q,hq,s,hs,hz⟩ |
| refine ⟨q,hq,s,?_,hz⟩ |
| rwa [aperture_eq_Icc τ η b q hτ hη hb (abs_le.mpr (interval_subset_unit hnodes hj hq))] |
| · rintro ⟨q,hq,s,hs,hz⟩ |
| refine ⟨q,hq,s,?_,hz⟩ |
| rw [aperture_eq_Icc τ η b q hτ hη hb (abs_le.mpr (interval_subset_unit hnodes hj hq))] at hs |
| exact hs |
|
|
| end CylinderCovering.CartesianCaps |
|
|
| end OAI |
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|
