File size: 3,968 Bytes
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 | import OAI.Geometry.IsometricImmersion.Comparison.QFirstJetComparison
import OAI.Geometry.IsometricImmersion.Taylor.TaylorCauchyData
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Taylor
theorem exists_reference_XX_tolerance (B : ℝ) {d c : ℝ} (hd : 0 < d) (hc : 0 < c) :
∃ epsilon : ℝ, 0 < epsilon ∧ ∀ a b : QFirstInput,
‖a‖ ≤ B → d ≤ qFirstDet a → c ≤ |qFirstXX a| →
‖b - a‖ ≤ epsilon → c / 2 ≤ |qFirstXX b| := by
obtain ⟨H, _, _, hH, _, _, hcompare⟩ := exists_uniform_qFirst_comparison B hd hc
let D := 4 * max B 0 + 2
have hD : 0 < D := by dsimp [D]; positivity
let epsilon := min 1 (min (d / (2 * D)) (c / (2 * (H + 1))))
have he : 0 < epsilon := by dsimp [epsilon]; positivity
refine ⟨epsilon, he, ?_⟩
intro a b ha hdet hxx hdist
have he1 : epsilon ≤ 1 := min_le_left _ _
have hed : epsilon ≤ d / (2 * D) := (min_le_right _ _).trans (min_le_left _ _)
have hec : epsilon ≤ c / (2 * (H + 1)) := (min_le_right _ _).trans (min_le_right _ _)
have hdist0 : 0 ≤ ‖b - a‖ := norm_nonneg _
have hdsmall : D * ‖b - a‖ ≤ d / 2 := by
have hh := (le_div_iff₀ (show 0 < 2 * D by positivity)).mp (hdist.trans hed)
nlinarith
have hcsmall : H * ‖b - a‖ ≤ c / 2 := by
have hh := (le_div_iff₀ (show 0 < 2 * (H + 1) by positivity)).mp (hdist.trans hec)
nlinarith
exact (hcompare a b ha hdet hxx (hdist.trans he1) hdsmall hcsmall).1 b
(right_mem_segment ℝ a b)
theorem exists_actual_reference_XX_tolerance (G W : ℝ) {d c : ℝ} (hd : 0 < d) (hc : 0 < c) :
∃ epsilon : ℝ, 0 < epsilon ∧ ∀ (g h : MetricField) (w : DarbouxState),
‖actualCurvatureFirstInput g (statePoint w)‖ ≤ G → ‖w‖ ≤ W →
d ≤ (g (statePoint w)).det → c ≤ |stateQDenominator g w| →
‖actualCurvatureFirstInput h (statePoint w) - actualCurvatureFirstInput g (statePoint w)‖ ≤ epsilon →
c / 2 ≤ |stateQDenominator h w| := by
obtain ⟨epsilon, he, htol⟩ := exists_reference_XX_tolerance (max G W) hd hc
refine ⟨epsilon, he, ?_⟩
intro g h w hg hw hdet hxx hdist
have ha : ‖qFirstBundle g w‖ ≤ max G W := by
change max ‖actualCurvatureFirstInput g (statePoint w)‖ ‖w‖ ≤ max G W
exact max_le_max hg hw
exact htol (qFirstBundle g w) (qFirstBundle h w) ha hdet hxx
(by simpa only [qFirstBundle_distance] using hdist)
theorem exists_reference_linearCauchy_XX_tolerance (G W : ℝ) {d c : ℝ}
(hd : 0 < d) (hc : 0 < c) :
∃ epsilon : ℝ, 0 < epsilon ∧ ∀ (g h : MetricField) (z : Coord → ℝ) (U : Set Coord),
ContDiffOn ℝ ∞ z U → IsOpen U → ∀ left right a : ℝ,
(∀ x ∈ Ioo left right, (![x, a] : Coord) ∈ U) →
∀ x ∈ Ioo left right,
‖actualCurvatureFirstInput g ![x, a]‖ ≤ G → ‖qSolutionJet z ![x, a]‖ ≤ W →
d ≤ (g ![x, a]).det → c ≤ |covHessian g z ![x, a] 0 0| →
‖actualCurvatureFirstInput h ![x, a] - actualCurvatureFirstInput g ![x, a]‖ ≤ epsilon →
c / 2 ≤ |covHessian h (linearCauchy a (heightCauchyValue z a) (heightCauchyVelocity z a))
![x, a] 0 0| := by
obtain ⟨epsilon, he, htol⟩ := exists_actual_reference_XX_tolerance G W hd hc
refine ⟨epsilon, he, ?_⟩
intro g h z U hz hU left right a hcut x hx hg hw hdet hxx hdist
rw [linearCauchy_height_initial_hessian hz hU left right a hcut hx]
have hb := htol g h (qSolutionJet z ![x, a])
(by simpa only [statePoint_qSolutionJet] using hg) hw
(by simpa only [statePoint_qSolutionJet] using hdet)
(by simpa only [stateQDenominator_qSolutionJet] using hxx)
(by simpa only [statePoint_qSolutionJet] using hdist)
simpa only [stateQDenominator_qSolutionJet] using hb
end SmoothLocal.Pulse
end
end OAI
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