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785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | import OAI.Geometry.CylinderCovering.Cartesian.GeneralDefinitions
import OAI.Geometry.CylinderCovering.Cartesian.Caps
namespace OAI
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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