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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import OAI.Geometry.IsometricImmersion.Comparison.ClosureMetricApproximation
import OAI.Geometry.IsometricImmersion.Pulses.PulseNeighborhoodAdapters
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise Distributions
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.Perturbation
def metricApproximationAccuracy (tau : ℕ) : ℝ := 1 / ((tau : ℝ)^tau)
theorem metricApproximationAccuracy_pos {tau : ℕ} (htau : 0 < tau) :
0 < metricApproximationAccuracy tau :=
one_div_pos.mpr (pow_pos (Nat.cast_pos.mpr htau) tau)
theorem relative_closure_exists_metric_approximation
{g0 : MetricField} {U : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 U)
{A : Set (metricPatchSet g0 kappa)} {theta : metricPatchSet g0 kappa}
(hclosure : theta ∈ closure A) (n : ℕ) {epsilon : ℝ} (hepsilon : 0 < epsilon) :
∃ eta : metricPatchSet g0 kappa, eta ∈ A ∧
SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧
(∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 eta.val) p < -kappa / 2) ∧
(∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x -
perturbedMetric g0 theta.val x) p‖ ≤ epsilon) ∧
(∀ i j : Fin 2, ∀ k ≤ n, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j -
perturbedMetric g0 theta.val x i j) p‖ ≤ epsilon) := by
obtain ⟨eta, heta, hT, hC⟩ := closure_exists_metric_jet_approximation
(Subtype.val : metricPatchSet g0 kappa → SymmetricPerturbation) continuous_subtype_val
g0 hclosure n hepsilon
exact ⟨eta, heta, perturbedMetric_smoothPositiveOn hg0 eta.val eta.property.1,
eta.property.2, hT, hC⟩
theorem exists_N_before_delta_eventually_class_approximation
{g0 : MetricField} {U : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
{A O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O)
{etaStar : metricPatchSet g0 kappa} (hStar : etaStar ∈ O)
(hOclosure : O ⊆ closure A)
(q0 a : ℝ) (ha : 0 < a) (hax : a ≤ 1 / 10) (hq : |q0| * a < 1 / 10) :
∃ N : ℕ, 10 < N ∧ ∀ delta : ℝ, 0 < delta →
∀ᶠ tau : ℕ in atTop, 0 < tau ∧
∃ eta : metricPatchSet g0 kappa, eta ∈ A ∧
SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧
(∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 eta.val) p < -kappa / 2) ∧
(∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x) p‖ ≤
metricApproximationAccuracy tau) ∧
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x i j) p‖ ≤
metricApproximationAccuracy tau) := by
obtain ⟨N, hN, hpulse⟩ := exists_N_before_delta_eventually_relative_metric_neighborhood
hg0 hU hSU hO hStar q0 a ha hax hq
refine ⟨N, hN, ?_⟩
intro delta hdelta
filter_upwards [hpulse delta hdelta] with tau htau
have hpositive : 0 < tau := by
have hh : (0 : ℝ) < (tau : ℝ) := zero_lt_one.trans_le htau.1
exact_mod_cast hh
obtain ⟨theta, hthetaO, hthetaEq, hthetaPositive, hthetaCentral⟩ := htau.2
obtain ⟨eta, heta, hmetric, hcentral, hT, hC⟩ :=
relative_closure_exists_metric_approximation hg0 (hOclosure hthetaO) tau
(metricApproximationAccuracy_pos hpositive)
refine ⟨hpositive, eta, heta, hmetric, hcentral, ?_, ?_⟩
· simpa only [hthetaEq] using hT
· simpa only [hthetaEq] using hC
theorem exists_N_before_delta_class_metric_family
{g0 : MetricField} {U : Set Coord} {kappa : ℝ}
(hg0 : SmoothPositiveOn g0 U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
{A O : Set (metricPatchSet g0 kappa)} (hO : IsOpen O)
{etaStar : metricPatchSet g0 kappa} (hStar : etaStar ∈ O)
(hOclosure : O ⊆ closure A)
(q0 a : ℝ) (ha : 0 < a) (hax : a ≤ 1 / 10) (hq : |q0| * a < 1 / 10) :
∃ N : ℕ, 10 < N ∧ ∀ delta : ℝ, 0 < delta →
∃ tau0 : ℕ, 1 ≤ tau0 ∧ ∃ eta : ℕ → metricPatchSet g0 kappa,
∀ tau : ℕ, tau0 ≤ tau → eta tau ∈ A ∧
SmoothPositiveOn (perturbedMetric g0 (eta tau).val) U ∧
(∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 (eta tau).val) p < -kappa / 2) ∧
(∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (eta tau).val x -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x) p‖ ≤
metricApproximationAccuracy tau) ∧
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 (eta tau).val x i j -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x i j) p‖ ≤
metricApproximationAccuracy tau) := by
classical
obtain ⟨N, hN, happrox⟩ := exists_N_before_delta_eventually_class_approximation
hg0 hU hSU hO hStar hOclosure q0 a ha hax hq
refine ⟨N, hN, ?_⟩
intro delta hdelta
obtain ⟨tauBase, hbase⟩ := eventually_atTop.mp (happrox delta hdelta)
let tau0 := max 1 tauBase
have hchoose : ∀ tau : ℕ, ∃ eta : metricPatchSet g0 kappa,
tau0 ≤ tau → eta ∈ A ∧ SmoothPositiveOn (perturbedMetric g0 eta.val) U ∧
(∀ p ∈ centralBox, gaussianCurvature (perturbedMetric g0 eta.val) p < -kappa / 2) ∧
(∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x) p‖ ≤
metricApproximationAccuracy tau) ∧
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p : Coord,
‖iteratedFDeriv ℝ k (fun x => perturbedMetric g0 eta.val x i j -
testMetric (perturbedMetric g0 etaStar.val) q0 a N delta (tau : ℝ) x i j) p‖ ≤
metricApproximationAccuracy tau) := by
intro tau
by_cases htau : tau0 ≤ tau
· obtain ⟨eta, heta⟩ := (hbase tau ((le_max_right 1 tauBase).trans htau)).2
exact ⟨eta, fun _ => heta⟩
· exact ⟨etaStar, fun h => False.elim (htau h)⟩
choose eta heta using hchoose
exact ⟨tau0, le_max_left _ _, eta, heta⟩
end SmoothLocal.Pulse
end
end OAI
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