lean/OAI/Geometry/IsometricImmersion/Comparison/FiniteApproximationJets.lean

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