File size: 3,568 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 | import OAI.Geometry.IsometricImmersion.Metrics.FixedMetricRemainder
import OAI.Geometry.IsometricImmersion.Comparison.PulseClosureApproximation
import OAI.Geometry.IsometricImmersion.Metrics.MetricLocality
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE SmoothLocal.HighEquation
theorem fixed_metric_finite_coefficient_jet_bound
{g : MetricField} {U : Set Coord} (hg : SmoothPositiveOn g U)
(hU : IsOpen U) (hSU : modelSquare ⊆ U) (m : ℕ) :
∃ B : ℝ, 0 ≤ B ∧ ∀ i j : Fin 2, ∀ k ≤ m, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ B := by
classical
have hco : ∀ i j : Fin 2, ∃ A : ℝ, 0 ≤ A ∧
∀ k ≤ m, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k (fun q => g q i j) p‖ ≤ A :=
fun i j => compact_finite_fullJet_bound (hg.1 i j) hU modelSquare_isCompact hSU m
choose A hA0 hA using hco
refine ⟨∑ i : Fin 2, ∑ j : Fin 2, A i j,
Finset.sum_nonneg (fun i _ => Finset.sum_nonneg (fun j _ => hA0 i j)),?_⟩
intro i j k hk p hp
exact (hA i j k hk p hp).trans
((Finset.single_le_sum (fun j _ => hA0 i j) (Finset.mem_univ j)).trans
(Finset.single_le_sum (fun a _ => Finset.sum_nonneg (fun b _ => hA0 a b))
(Finset.mem_univ i)))
theorem metricApproximationAccuracy_le_one {tau : ℕ} (htau : 1 ≤ tau) :
metricApproximationAccuracy tau ≤ 1 := by
have htauR : (1 : ℝ) ≤ (tau : ℝ) := by exact_mod_cast htau
simpa only [metricApproximationAccuracy,div_one] using
(one_div_le_one_div_of_le (by norm_num : (0 : ℝ)<1) (one_le_pow₀ htauR : 1 ≤ (tau : ℝ)^tau))
theorem metric_jet_norm_le_of_approximation_locality
{gStar gTau test : MetricField} {U V : Set Coord}
(hgStar : SmoothPositiveOn gStar U) (hgTau : SmoothPositiveOn gTau V)
(hU : IsOpen U) (hV : IsOpen V) {p : Coord} (hpU : p ∈ U) (hpV : p ∈ V)
(heq : test =ᶠ[𝓝 p] gStar) (i j : Fin 2) (k : ℕ) {B epsilon : ℝ}
(hbase : ‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ B)
(happrox : ‖iteratedFDeriv ℝ k (fun q => gTau q i j-test q i j) p‖ ≤ epsilon) :
‖iteratedFDeriv ℝ k (fun q => gTau q i j) p‖ ≤ B+epsilon := by
have heqCoeff : (fun q => gTau q i j-test q i j) =ᶠ[𝓝 p]
(fun q => gTau q i j-gStar q i j) := by
filter_upwards [metric_coeff_eventuallyEq heq i j] with q hq
rw [hq]
have hdiff : ‖iteratedFDeriv ℝ k (fun q => gTau q i j-gStar q i j) p‖ ≤ epsilon := by
rw [←(heqCoeff.iteratedFDeriv ℝ k).self_of_nhds]
exact happrox
have hsplit := iteratedFDeriv_sub_apply (i := k)
(((hgTau.1 i j).contDiffAt (hV.mem_nhds hpV)).of_le (WithTop.coe_le_coe.mpr le_top))
(((hgStar.1 i j).contDiffAt (hU.mem_nhds hpU)).of_le (WithTop.coe_le_coe.mpr le_top))
change ‖iteratedFDeriv ℝ k ((fun q => gTau q i j)-(fun q => gStar q i j)) p‖ ≤ epsilon at hdiff
rw [hsplit] at hdiff
calc
_ = ‖(iteratedFDeriv ℝ k (fun q => gTau q i j) p-
iteratedFDeriv ℝ k (fun q => gStar q i j) p)+
iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ := by rw [sub_add_cancel]
_ ≤ ‖iteratedFDeriv ℝ k (fun q => gTau q i j) p-
iteratedFDeriv ℝ k (fun q => gStar q i j) p‖+
‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ := norm_add_le _ _
_ ≤ epsilon+B := add_le_add hdiff hbase
_ = B+epsilon := add_comm _ _
end SmoothLocal.Pulse
end
end OAI
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