| 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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