| import OAI.Geometry.IsometricImmersion.Darboux.QFirstJetFunctions | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped ContDiff Topology BigOperators Matrix Matrix.Norms.Elementwise | |
| namespace SmoothLocal.Pulse | |
| open SmoothLocal.Geometry SmoothLocal.HighEquation | |
| theorem qFirst_segment_distance_le (a b : QFirstInput) {x : QFirstInput} | |
| (hx : x ∈ segment ℝ a b) : ‖x-a‖ ≤ ‖b-a‖ := by | |
| have hball : x ∈ Metric.closedBall a ‖b-a‖ := | |
| (convex_closedBall a ‖b-a‖).segment_subset | |
| (by simp) (by simp only [Metric.mem_closedBall,dist_eq_norm,le_refl]) hx | |
| simpa only [Metric.mem_closedBall,dist_eq_norm] using hball | |
| theorem qFirst_close_segment_subset_baseTube (a b : QFirstInput) {B d : ℝ} | |
| (ha : ‖a‖ ≤ B) (hdet : d ≤ qFirstDet a) (hdist : ‖b-a‖ ≤ 1) | |
| (hsmall : (4*max B 0+2)*‖b-a‖ ≤ d/2) : | |
| segment ℝ a b ⊆ qFirstBaseTube (max B 0+1) (d/2) := by | |
| intro x hx | |
| have hxa := qFirst_segment_distance_le a b hx | |
| have hxnorm : ‖x‖ ≤ max B 0+1 := by | |
| calc | |
| _ = ‖(x-a)+a‖ := by rw [sub_add_cancel] | |
| _ ≤ ‖x-a‖+‖a‖ := norm_add_le _ _ | |
| _ ≤ 1+max B 0 := add_le_add (hxa.trans hdist) (ha.trans (le_max_left _ _)) | |
| _ = _ := by ring | |
| have hmatrixD : ‖x.1.1-a.1.1‖ ≤ ‖b-a‖ := | |
| (show ‖x.1.1-a.1.1‖ ≤ ‖x-a‖ from | |
| (norm_fst_le (x-a).1).trans (norm_fst_le (x-a))).trans hxa | |
| have hmatrixA : ‖a.1.1‖ ≤ max B 0 := | |
| ((norm_fst_le a.1).trans (norm_fst_le a)).trans (ha.trans (le_max_left _ _)) | |
| have hdetdiff := det_sub_le_small_first_distance (Matrix.of a.1.1) (Matrix.of x.1.1) | |
| (le_max_right B 0) (norm_nonneg _) hdist hmatrixA hmatrixD | |
| have hxdet : d/2 ≤ qFirstDet x := by | |
| have hneg := neg_abs_le (qFirstDet x-qFirstDet a) | |
| change |qFirstDet x-qFirstDet a| ≤ (4*max B 0+2)*‖b-a‖ at hdetdiff | |
| linarith [hdetdiff.trans hsmall] | |
| exact ⟨by simpa only [Metric.mem_closedBall,dist_zero_right] using hxnorm,hxdet⟩ | |
| theorem exists_uniform_qFirst_comparison (B : ℝ) {d c : ℝ} | |
| (hd : 0 < d) (hc : 0 < c) : | |
| ∃ H F L : ℝ, 0 ≤ H ∧ 0 ≤ F ∧ 0 ≤ L ∧ | |
| ∀ a b : QFirstInput, ‖a‖ ≤ B → d ≤ qFirstDet a → c ≤ |qFirstXX a| → | |
| ‖b-a‖ ≤ 1 → (4*max B 0+2)*‖b-a‖ ≤ d/2 → H*‖b-a‖ ≤ c/2 → | |
| (∀ x ∈ segment ℝ a b, c/2 ≤ |qFirstXX x|) ∧ | |
| |qFirstDensityFactor b-qFirstDensityFactor a| ≤ F*‖b-a‖ ∧ | |
| |qFirstLower b-qFirstLower a| ≤ L*‖b-a‖ := by | |
| obtain ⟨H,hH,hHb⟩ := qFirstXX_fderiv_bound (max B 0+1) (half_pos hd) | |
| obtain ⟨F,hF,hFb⟩ := qFirstTube_fderiv_bound qFirstDensityFactor | |
| qFirstDensityFactor_contDiffOn (max B 0+1) (half_pos hd) (half_pos hc) | |
| obtain ⟨L,hL,hLb⟩ := qFirstTube_fderiv_bound qFirstLower | |
| qFirstLower_contDiffOn (max B 0+1) (half_pos hd) (half_pos hc) | |
| refine ⟨H,F,L,hH,hF,hL,?_⟩ | |
| intro a b ha hdet hxx hdist hsmall hxxsmall | |
| have hbase := qFirst_close_segment_subset_baseTube a b ha hdet hdist hsmall | |
| have hXXdiff : ∀ x ∈ segment ℝ a b, DifferentiableAt ℝ qFirstXX x := by | |
| intro x hx | |
| exact (qFirstXX_contDiffOn.contDiffAt (qFirstBase_isOpen.mem_nhds | |
| (qFirstBaseTube_subset_base (max B 0+1) (half_pos hd) (hbase hx)))).differentiableAt (by simp) | |
| have hfloor : ∀ x ∈ segment ℝ a b, c/2 ≤ |qFirstXX x| := by | |
| intro x hx | |
| have herr : |qFirstXX x-qFirstXX a| ≤ H*‖x-a‖ := by | |
| simpa only [Real.norm_eq_abs] using | |
| (convex_segment a b).norm_image_sub_le_of_norm_fderiv_le hXXdiff | |
| (fun y hy => hHb y (hbase hy)) (left_mem_segment ℝ a b) hx | |
| have hsmall' : |qFirstXX x-qFirstXX a| ≤ c/2 := | |
| herr.trans ((mul_le_mul_of_nonneg_left (qFirst_segment_distance_le a b hx) hH).trans hxxsmall) | |
| have htriangle : |qFirstXX a| ≤ |qFirstXX x|+|qFirstXX x-qFirstXX a| := by | |
| calc | |
| _ = |qFirstXX x-(qFirstXX x-qFirstXX a)| := by congr 1; ring | |
| _ ≤ _ := abs_sub _ _ | |
| linarith | |
| have htube : segment ℝ a b ⊆ qFirstTube (max B 0+1) (d/2) (c/2) := | |
| fun x hx => ⟨hbase hx,hfloor x hx⟩ | |
| have hcompare (f : QFirstInput → ℝ) (hf : ContDiffOn ℝ ∞ f qFirstDomain) | |
| (C : ℝ) (hCb : ∀ x ∈ qFirstTube (max B 0+1) (d/2) (c/2), ‖fderiv ℝ f x‖ ≤ C) : | |
| |f b-f a| ≤ C*‖b-a‖ := by | |
| have hdiff : ∀ x ∈ segment ℝ a b, DifferentiableAt ℝ f x := by | |
| intro x hx | |
| exact (hf.contDiffAt (qFirstDomain_isOpen.mem_nhds | |
| (qFirstTube_subset_domain (max B 0+1) (half_pos hd) (half_pos hc) | |
| (htube hx)))).differentiableAt (by simp) | |
| simpa only [Real.norm_eq_abs] using | |
| (convex_segment a b).norm_image_sub_le_of_norm_fderiv_le hdiff | |
| (fun x hx => hCb x (htube hx)) (left_mem_segment ℝ a b) (right_mem_segment ℝ a b) | |
| exact ⟨hfloor,hcompare _ qFirstDensityFactor_contDiffOn F hFb, | |
| hcompare _ qFirstLower_contDiffOn L hLb⟩ | |
| end SmoothLocal.Pulse | |
| end | |
| end OAI | |