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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | import OAI.Geometry.IsometricImmersion.Pulses.PulseDensitySlab
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry
theorem matrix_entry_abs_le_norm (G : MetricMatrix) (i j : Fin 2) : |G i j| ≤ ‖G‖ := by
simpa only [Real.norm_eq_abs] using
(Matrix.norm_entry_le_entrywise_sup_norm G (i := i) (j := j))
theorem det_sub_le_first_distance (G H : MetricMatrix) {B e : ℝ}
(hB : 0 ≤ B) (he : 0 ≤ e) (hG : ‖G‖ ≤ B) (hD : ‖H-G‖ ≤ e) :
|H.det-G.det| ≤ 4*B*e+2*e^2 := by
let D := H-G
have hg (i j : Fin 2) : |G i j| ≤ B := (matrix_entry_abs_le_norm G i j).trans hG
have hd (i j : Fin 2) : |D i j| ≤ e := (matrix_entry_abs_le_norm D i j).trans hD
have hgd (i j k l : Fin 2) : |G i j*D k l| ≤ B*e := by
rw [abs_mul]
exact mul_le_mul (hg i j) (hd k l) (abs_nonneg _) hB
have hdg (i j k l : Fin 2) : |D i j*G k l| ≤ e*B := by
rw [abs_mul]
exact mul_le_mul (hd i j) (hg k l) (abs_nonneg _) he
have hdd (i j k l : Fin 2) : |D i j*D k l| ≤ e^2 := by
rw [abs_mul,pow_two]
exact mul_le_mul (hd i j) (hd k l) (abs_nonneg _) he
have hsum (i j k l : Fin 2) :
|G i j*D k l+D i j*G k l+D i j*D k l| ≤ B*e+e*B+e^2 :=
((abs_add_le _ _).trans (add_le_add (abs_add_le _ _) le_rfl)).trans
(add_le_add (add_le_add (hgd i j k l) (hdg i j k l)) (hdd i j k l))
have hEq : H.det-G.det =
(G 0 0*D 1 1+D 0 0*G 1 1+D 0 0*D 1 1)-
(G 0 1*D 1 0+D 0 1*G 1 0+D 0 1*D 1 0) := by
simp only [D,Matrix.det_fin_two,Matrix.sub_apply]
ring
rw [hEq]
calc
_ ≤ |G 0 0*D 1 1+D 0 0*G 1 1+D 0 0*D 1 1|+
|G 0 1*D 1 0+D 0 1*G 1 0+D 0 1*D 1 0| := abs_sub _ _
_ ≤ (B*e+e*B+e^2)+(B*e+e*B+e^2) := add_le_add (hsum 0 0 1 1) (hsum 0 1 1 0)
_ = _ := by ring
theorem det_sub_le_small_first_distance (G H : MetricMatrix) {B e : ℝ}
(hB : 0 ≤ B) (he : 0 ≤ e) (he1 : e ≤ 1) (hG : ‖G‖ ≤ B) (hD : ‖H-G‖ ≤ e) :
|H.det-G.det| ≤ (4*B+2)*e := by
apply (det_sub_le_first_distance G H hB he hG hD).trans
have hee : e^2 ≤ e := by nlinarith [mul_le_mul_of_nonneg_left he1 he]
nlinarith
theorem first_jet_close_segment_subset_tube (a b : CurvatureFirstInput) {B d : ℝ}
(ha : ‖a‖ ≤ B) (hdet : d ≤ (Matrix.of a.1).det) (hdist : ‖b-a‖ ≤ 1)
(hsmall : (4*max B 0+2)*‖b-a‖ ≤ d/2) :
segment ℝ a b ⊆ curvatureFirstTube (max B 0+1) (d/2) := by
intro x hx
have hxa : ‖x-a‖ ≤ ‖b-a‖ := by
have hball : x ∈ Metric.closedBall a ‖b-a‖ :=
(convex_closedBall a ‖b-a‖).segment_subset
(by simp) (Metric.mem_closedBall.mpr (dist_eq_norm b a).le) hx
simpa only [Metric.mem_closedBall,dist_eq_norm] using hball
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-a.1‖ ≤ ‖b-a‖ :=
(show ‖x.1-a.1‖ ≤ ‖x-a‖ from norm_fst_le (x-a)).trans hxa
have hmatrixA : ‖a.1‖ ≤ max B 0 := (norm_fst_le a).trans (ha.trans (le_max_left _ _))
have hdetdiff := det_sub_le_small_first_distance (Matrix.of a.1) (Matrix.of x.1) (le_max_right B 0)
(norm_nonneg _) hdist hmatrixA hmatrixD
have hxdet : d/2 ≤ (Matrix.of x.1).det := by
have hneg := neg_abs_le ((Matrix.of x.1).det-(Matrix.of a.1).det)
linarith [hdetdiff.trans hsmall]
exact ⟨by simpa only [Metric.mem_closedBall,dist_zero_right] using hxnorm,hxdet⟩
theorem exists_uniform_curvature_first_comparison (B : ℝ) {d : ℝ} (hd : 0 < d) :
∃ C : ℝ, 0 ≤ C ∧ ∀ a b : CurvatureFirstInput,
‖a‖ ≤ B → d ≤ (Matrix.of a.1).det → ‖b-a‖ ≤ 1 →
(4*max B 0+2)*‖b-a‖ ≤ d/2 →
|curvatureFirstFunction b-curvatureFirstFunction a| ≤ C*‖b-a‖ := by
obtain ⟨C,hC,hbound⟩ := curvatureFirstTube_fderiv_bound (max B 0+1) (half_pos hd)
refine ⟨C,hC,?_⟩
intro a b ha hdet hdist hsmall
have htube := first_jet_close_segment_subset_tube a b ha hdet hdist hsmall
have hdiff : ∀ x ∈ segment ℝ a b, DifferentiableAt ℝ curvatureFirstFunction x := by
intro x hx
exact (curvatureFirstFunction_contDiffOn.contDiffAt (curvatureFirstDomain_isOpen.mem_nhds
(curvatureFirstTube_subset_domain (max B 0+1) (half_pos hd) (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 => hbound x (htube hx)) (left_mem_segment ℝ a b) (right_mem_segment ℝ a b)
theorem exists_uniform_curvature_density_comparison (B : ℝ) {d : ℝ} (hd : 0 < d) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (g h : MetricField) (U : Set Coord),
SmoothPositiveOn g U → SmoothPositiveOn h U → IsOpen U →
∀ p ∈ U, ∀ e : ℝ,
‖actualCurvatureFirstInput h p‖ ≤ B → d ≤ (h p).det →
‖actualCurvatureFirstInput g p-actualCurvatureFirstInput h p‖ ≤ e →
e ≤ 1 → (4*max B 0+2)*e ≤ d/2 →
(∀ i j k l : Fin 2,
|coordPartial i (coordPartial j (fun q => g q k l)) p-
coordPartial i (coordPartial j (fun q => h q k l)) p| ≤ e) →
|curvatureDensity g p-curvatureDensity h p| ≤ C*e := by
obtain ⟨C0,hC0,hbound⟩ := exists_uniform_curvature_first_comparison B hd
refine ⟨2+C0,by linarith,?_⟩
intro g h U hg hh hU p hp e hjet hdet hdist he1 hsmall hsecond
have hscale : 0 ≤ 4*max B 0+2 := by linarith [le_max_right B 0]
have hfirst := hbound (actualCurvatureFirstInput h p) (actualCurvatureFirstInput g p)
hjet hdet (hdist.trans he1) ((mul_le_mul_of_nonneg_left hdist hscale).trans hsmall)
have hL : |curvatureLowerTerm g p-curvatureLowerTerm h p| ≤ C0*e :=
hfirst.trans (mul_le_mul_of_nonneg_left hdist hC0)
rw [←curvatureDensityJetError_eq hg hh hU hp]
exact ((curvatureDensityJetError_le g h p (hsecond 0 1 0 1) (hsecond 0 0 1 1)
(hsecond 1 1 0 0)).trans (add_le_add le_rfl hL)).trans_eq (by ring)
end SmoothLocal.Pulse
end
end OAI
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