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59d1c80 | 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 | 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
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