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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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Geometry.CylinderCovering.Ruled.LocalDefinitions
import OAI.Geometry.CylinderCovering.Ruled.CellGeometry
import OAI.Geometry.CylinderCovering.Convexity.CompactC1Remainder
namespace OAI
noncomputable section
open Set Metric
namespace CylinderCovering.ProductLocal
open RuledApproximation
lemma frame_apply (L : (ℝ × ℝ) ≃L[ℝ] Plane) (s : ℝ × ℝ) :
L s = s.1 • L (1,0) + s.2 • L (0,1) := by
rw [← map_smul, ← map_smul, ← map_add]
congr 1
ext <;> simp
lemma frame_norm_le {A : Plane →L[ℝ] Plane} {lam : ℝ}
{L : (ℝ × ℝ) ≃L[ℝ] Plane} (hL : IsUnitEigenframe A lam L)
(s : ℝ × ℝ) : ‖L s‖ ≤ |s.1| + |s.2| := by
rw [frame_apply]
calc ‖s.1 • L (1,0) + s.2 • L (0,1)‖
≤ ‖s.1 • L (1,0)‖ + ‖s.2 • L (0,1)‖ := norm_add_le _ _
_ = |s.1| + |s.2| := by rw [norm_smul,norm_smul,hL.1,hL.2.1]; simp
lemma frame_diagonal {A : Plane →L[ℝ] Plane} {lam : ℝ}
{L : (ℝ × ℝ) ≃L[ℝ] Plane} (hL : IsUnitEigenframe A lam L)
(s : ℝ × ℝ) : A (L s) = L (lam*s.1,-lam*s.2) := by
rw [frame_apply L s,map_add,map_smul,map_smul,hL.2.2.1,hL.2.2.2,
frame_apply L (lam*s.1,-lam*s.2)]
simp only [smul_smul]
congr 1 <;> congr 1 <;> ring
lemma affineCell_distance {A : Plane →L[ℝ] Plane} {lam tau rho : ℝ}
{L : (ℝ × ℝ) ≃L[ℝ] Plane} (hL : IsUnitEigenframe A lam L)
{p q : Plane} (hrho : 0 ≤ rho)
(hq : q ∈ affineCell p rho L (ProductCells.parameterCell tau)) :
‖q-p‖ ≤ 2*rho := by
obtain ⟨s,hs,rfl⟩ := hq
simp only [add_sub_cancel_left,norm_smul,Real.norm_eq_abs,abs_of_nonneg hrho]
have hb : ‖L s‖ ≤ 2 := (frame_norm_le hL s).trans (by linarith [hs.1,hs.2.1])
nlinarith
lemma intercept_identity (v : Plane → Plane) (p : Plane) (rho x lam : ℝ)
(L : (ℝ × ℝ) ≃L[ℝ] Plane)
(hL : IsUnitEigenframe (fderiv ℝ v p) lam L) (hrho : rho ≠ 0)
(s : ℝ × ℝ) :
let q := p + rho • L s
let e := (x/rho) • L.symm (v q-v p-fderiv ℝ v p (q-p))
q + x • (v q-v p) =
p + rho • L (((1+x*lam)*s.1+e.1,(1-x*lam)*s.2+e.2)) := by
dsimp only
have hpair : (((1+x*lam)*s.1, (1-x*lam)*s.2) : ℝ × ℝ) =
s + x • (lam*s.1,-lam*s.2) := by ext <;> simp <;> ring
have hepair (e : ℝ × ℝ) :
((1+x*lam)*s.1+e.1,(1-x*lam)*s.2+e.2) =
((1+x*lam)*s.1,(1-x*lam)*s.2)+e := rfl
rw [hepair,hpair,map_add,map_add,map_smul,← frame_diagonal hL,
map_smul,L.apply_symm_apply,smul_add,smul_add]
simp only [smul_smul]
rw [show rho*(x/rho)=x by field_simp]
simp only [add_sub_cancel_left,map_smul,smul_sub,smul_smul]
module
theorem uniform_intercept_containment
{D U : Set Plane} (hD : IsCompact D) (hU : IsOpen U) (hDU : D ⊆ U)
{v : Plane → Plane} (hv : ContDiffOn ℝ 1 v U)
{M N tau gamma : ℝ} (hM : 0 < M) (hN : 0 < N)
(htau : 0 < tau) (hgamma : 0 < gamma) :
∃ rho0 : ℝ, 0 < rho0 ∧ ∀ p ∈ D,
∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane),
IsUnitEigenframe (fderiv ℝ v p) lam L →
0 ≤ lam → M*lam ≤ 1 → (∀ z, ‖L.symm z‖ ≤ N*‖z‖) →
∀ rho : ℝ, 0 < rho → rho < rho0 →
∀ q ∈ affineCell p rho L (ProductCells.parameterCell tau),
∀ x : ℝ, |x| ≤ M →
q + x • (v q-v p) ∈ affineCell p rho L (ProductCells.interceptCell tau gamma) := by
let c : ℝ := min (1/2) (gamma*tau/5) / 2
have hc : 0 < c := by dsimp [c]; positivity
have hc1 : c < 1/2 := by dsimp [c]; linarith [min_le_left (1/2:ℝ) (gamma*tau/5)]
have hc2 : c < gamma*tau/5 := by
dsimp [c]; linarith [min_le_right (1/2:ℝ) (gamma*tau/5), mul_pos hgamma htau]
let delta : ℝ := c/(2*M*N)
have hd : 0 < delta := by dsimp [delta]; positivity
obtain ⟨r,hr,hrem⟩ := compact_c1_remainder hD hU hDU hv hd
refine ⟨r/3,by positivity,?_⟩
intro p hp lam L hL hlam hML hLN rho hrho hrhor q hq x hx
obtain ⟨s,hs,rfl⟩ := hq
let q := p+rho • L s
have hqdist : ‖q-p‖ ≤ 2*rho := affineCell_distance hL hrho.le ⟨s,hs,rfl⟩
have hqr : q ∈ ball p r := by rw [mem_ball,dist_eq_norm]; linarith
let w := v q-v p-fderiv ℝ v p (q-p)
have hw : ‖w‖ ≤ delta*(2*rho) :=
(hrem p hp).2 q hqr |>.trans (mul_le_mul_of_nonneg_left hqdist hd.le)
let e : ℝ × ℝ := (x/rho) • L.symm w
have he : ‖e‖ ≤ c := by
calc ‖e‖ = (|x|/rho) * ‖L.symm w‖ := by
simp only [e,norm_smul,Real.norm_eq_abs,abs_div,abs_of_pos hrho]
_ ≤ (M/rho) * (N*(delta*(2*rho))) := by
gcongr
exact (hLN w).trans (mul_le_mul_of_nonneg_left hw hN.le)
_ = c := by dsimp [delta]; field_simp
have he1 : |e.1| ≤ c := (show |e.1| ≤ ‖e‖ by simpa only [Real.norm_eq_abs] using norm_fst_le e).trans he
have he2 : |e.2| ≤ c := (show |e.2| ≤ ‖e‖ by simpa only [Real.norm_eq_abs] using norm_snd_le e).trans he
have hz : |x*lam| ≤ 1 := by
rw [abs_mul,abs_of_nonneg hlam]
exact (mul_le_mul_of_nonneg_right hx hlam).trans hML
refine ⟨((1+x*lam)*s.1+e.1,(1-x*lam)*s.2+e.2),
ProductCells.expanded_parameterCell_mem_intercept hz hs he1 he2 hc1 hc2,?_⟩
exact (intercept_identity v p rho x lam L hL hrho.ne' s).symm
end CylinderCovering.ProductLocal
end
end OAI
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