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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 | import OAI.Geometry.IsometricImmersion.Pulses.OffPulseMetricBounds
import OAI.Geometry.IsometricImmersion.Comparison.ShearFirstJetComparison
import OAI.Geometry.IsometricImmersion.Comparison.CurvatureFullJetComparison
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.HighEquation
theorem actual_sheared_reference_first_distance_of_locality
{gStar gTau test : MetricField} {U V : Set Coord}
(hgTau : SmoothPositiveOn gTau U) (hgStar : SmoothPositiveOn gStar V)
(hU : IsOpen U) (hV : IsOpen V) {q0 e : ℝ} (hq0 : |q0| ≤ 1) (he : 0 ≤ e)
{p : Coord} (hpU : inverseShearCoordinates q0 p ∈ U) (hpV : inverseShearCoordinates q0 p ∈ V)
(hlocal : test =ᶠ[𝓝 (inverseShearCoordinates q0 p)] gStar)
(happrox : ∀ i j : Fin 2, ∀ k ≤ 1,
‖iteratedFDeriv ℝ k (fun q => gTau q i j - test q i j) (inverseShearCoordinates q0 p)‖ ≤ e) :
‖actualCurvatureFirstInput (metricInShearCoordinates gStar q0) p -
actualCurvatureFirstInput (metricInShearCoordinates gTau q0) p‖ ≤ 8 * e := by
let W := U ∩ V
have hW : IsOpen W := hU.inter hV
have hgTW : SmoothPositiveOn gTau W :=
⟨fun i j => (hgTau.1 i j).mono inter_subset_left, fun q hq => hgTau.2 q hq.1⟩
have hgSW : SmoothPositiveOn gStar W :=
⟨fun i j => (hgStar.1 i j).mono inter_subset_right, fun q hq => hgStar.2 q hq.2⟩
have hpW : inverseShearCoordinates q0 p ∈ W := ⟨hpU, hpV⟩
have hdiff : ∀ i j : Fin 2, ∀ k ≤ 1,
‖iteratedFDeriv ℝ k (fun q => gTau q i j - gStar q i j) (inverseShearCoordinates q0 p)‖ ≤ e := by
intro i j k hk
have heq : (fun q => gTau q i j - test q i j) =ᶠ[𝓝 (inverseShearCoordinates q0 p)]
(fun q => gTau q i j - gStar q i j) := by
filter_upwards [metric_coeff_eventuallyEq hlocal i j] with q hq
rw [hq]
rw [← (heq.iteratedFDeriv ℝ k).self_of_nhds]
exact happrox i j k hk
have hfirst := actual_first_input_difference_le_full_jets hgTW hgSW hW hpW he hdiff
rw [norm_sub_rev]
exact actual_shear_first_jet_distance_le hgTW hgSW hW hq0 hpW hfirst
theorem metricApproximationAccuracy_le_inverse {tau : ℕ} (htau : 1 ≤ tau) :
metricApproximationAccuracy tau ≤ 1 / (tau : ℝ) := by
have htauR : (1 : ℝ) ≤ (tau : ℝ) := by exact_mod_cast htau
have htauP : (0 : ℝ) < (tau : ℝ) := zero_lt_one.trans_le htauR
unfold metricApproximationAccuracy
exact one_div_le_one_div_of_le htauP (le_self_pow₀ htauR (Nat.ne_of_gt htau))
theorem exists_first_error_threshold {epsilon : ℝ} (he : 0 < epsilon) :
∃ tau0 : ℕ, 1 ≤ tau0 ∧ ∀ tau : ℕ, tau0 ≤ tau →
8 * metricApproximationAccuracy tau ≤ epsilon := by
obtain ⟨n, hn⟩ := exists_nat_gt (8 / epsilon)
refine ⟨max 1 n, le_max_left _ _, ?_⟩
intro tau htau
have htauN : 1 ≤ tau := (le_max_left _ _).trans htau
have hnN : n ≤ tau := (le_max_right _ _).trans htau
have htauR : (0 : ℝ) < (tau : ℝ) := zero_lt_one.trans_le (by exact_mod_cast htauN)
have hnR : (n : ℝ) ≤ (tau : ℝ) := by exact_mod_cast hnN
have hh : 8 < (n : ℝ) * epsilon := (div_lt_iff₀ he).mp hn
have hb : 8 / (tau : ℝ) ≤ epsilon := (div_le_iff₀ htauR).mpr (by nlinarith)
exact (mul_le_mul_of_nonneg_left (metricApproximationAccuracy_le_inverse htauN) (by norm_num)).trans
(by simpa only [mul_one_div] using hb)
theorem actual_pulse_edge_reference_first_distance
{gStar gTau : MetricField} {U V : Set Coord}
(hgTau : SmoothPositiveOn gTau U) (hgStar : SmoothPositiveOn gStar V)
(hU : IsOpen U) (hV : IsOpen V) {q0 a delta : ℝ} {N tau : ℕ}
(hq0 : |q0| ≤ 1) (ha : 0 < a) (hd : 0 < delta) (htau : 1 ≤ tau)
{x : ℝ}
(hpU : inverseShearCoordinates q0 ![x, -delta / (tau : ℝ)] ∈ U)
(hpV : inverseShearCoordinates q0 ![x, -delta / (tau : ℝ)] ∈ V)
(happrox : ∀ i j : Fin 2, ∀ k ≤ tau,
‖iteratedFDeriv ℝ k
(fun q => gTau q i j - testMetric gStar q0 a N delta (tau : ℝ) q i j)
(inverseShearCoordinates q0 ![x, -delta / (tau : ℝ)])‖ ≤ metricApproximationAccuracy tau) :
‖actualCurvatureFirstInput (metricInShearCoordinates gStar q0) ![x, -delta / (tau : ℝ)] -
actualCurvatureFirstInput (metricInShearCoordinates gTau q0) ![x, -delta / (tau : ℝ)]‖ ≤
8 * metricApproximationAccuracy tau := by
have htauR : (0 : ℝ) < (tau : ℝ) := zero_lt_one.trans_le (by exact_mod_cast htau)
apply actual_sheared_reference_first_distance_of_locality hgTau hgStar hU hV hq0
(metricApproximationAccuracy_pos htau).le hpU hpV
(testMetric_eventuallyEq_below_pulse gStar q0 a N ha hd htauR le_rfl)
intro i j k hk
exact happrox i j k (hk.trans htau)
end SmoothLocal.Pulse
end
end OAI
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