import OAI.Geometry.IsometricImmersion.Estimates.ActualQApplicability import OAI.Geometry.IsometricImmersion.Comparison.QApproximationMargins import OAI.Geometry.IsometricImmersion.Comparison.ComparisonEnergyScaling namespace OAI noncomputable section open Set Filter Function MeasureTheory open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise namespace SmoothLocal.Perturbation open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation SmoothLocal.Flow open SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Hyperbolic SmoothLocal.Taylor theorem exists_actual_class_comparison_source_bound_at_radius {gStar : MetricField} {V : Set Coord} (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) {G d kappa q0 r : ℝ} (M : ℕ) (hG : 0 ≤ G) (hd : 0 < d) (hgStarB : ∀ i j k, k ≤ 2 → ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ G) (hdStar : ∀ p ∈ modelSquare, d ≤ (gStar p).det) (hkappa : 0 < kappa) (hM : 0 < M) (hq0 : |q0| ≤ 1/20) (hr : 0 < r) (hrhalf : r < 1/2) (hLr : boundedClassWidth kappa M*r ≤ 1/20) (hrsmall : heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))* (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M)) (N : ℕ) (hN : 2 < N) (Cres : ℝ) (hCres : 0 ≤ Cres) : ∃ Csource : ℝ, 0 < Csource ∧ ∀ delta : ℝ, 0 < delta → delta ≤ 1/2 → ∀ᶠ tau : ℕ in atTop, 1 ≤ tau ∧ ∀ (g0 : MetricField) (eta : metricPatchSet g0 kappa) (U W : Set Coord) (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ), SmoothPositiveOn (perturbedMetric g0 eta.val) U → IsOpen U → CapInductionHeight (perturbedMetric g0 eta.val) U (M : ℝ) (1/(M : ℝ)) (1/(M : ℝ)) z → CapInductionFlow (perturbedMetric g0 eta.val) U G (M : ℝ) d (1/(M : ℝ)) (1/(M : ℝ)) kappa z Y W → BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z → (∀ i j k, k ≤ 4 → ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j) p‖ ≤ G) → (∀ p ∈ modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) → |hessianQuotient (perturbedMetric g0 eta.val) z 0-q0| ≤ 1/(100*boundedClassWidth kappa M) → (∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare, ‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j- testMetric gStar q0 (boundedClassWidth kappa M*r/16) N delta (tau : ℝ) q i j) p‖ ≤ metricApproximationAccuracy tau) → ∀ P : Coord → ℝ, (∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ), |qResidual (metricInShearCoordinates gStar q0) P p| ≤ Cres/(tau : ℝ)^N) → ∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ), |actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P p| ≤ Csource*(tau : ℝ)^2/(tau : ℝ)^N := by let L := boundedClassWidth kappa M let a := L*r/16 let c := (boundedClassSpeed kappa M)^2/(4*(M : ℝ)) let BQ := max (8*G) (boundedClassShearedStateBudget q0 M) have ha : 0 < a := div_pos (mul_pos (boundedClassWidth_pos kappa M) hr) (by norm_num) have hc : 0 < c := by have hh := boundedClassShearHxxFloor_pos hkappa hM have he : c=((boundedClassSpeed kappa M)^2/(2*(M : ℝ)))/2 := by dsimp only [c]; ring rw [he] exact half_pos hh obtain ⟨_,A,D,dcompare,_,hA,hD,hdcompare,happlicable⟩ := exists_actual_Q_applicability_at_radius hgStar hV hSV M hG hd hgStarB hdStar hkappa hM hq0 hr hrhalf hLr hrsmall N hN obtain ⟨H,Cfirst,Ctest,Ccompare,_,hCf,_,hCc,hsource⟩ := exists_actual_comparison_source_pointwise_bound (8*G) (G+1) BQ D A Cres hd hdcompare hc hD hA ha N (by omega) let Csource := 2*A*pulseLeadingBound a ha+1 have hCs : 0 < Csource := by have hpulse := pulseLeadingBound_nonneg ha dsimp only [Csource] positivity refine ⟨Csource,hCs,?_⟩ intro delta hdelt hdhalf obtain ⟨T,_,hsourceT⟩ := hsource delta hdelt let Q := 8+scalarPulseFirstJetBound a ha delta have hlarge : ∀ᶠ tau : ℕ in atTop, T ≤ (tau : ℝ) ∧ comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta ≤ (tau : ℝ) := by apply (tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually (p := fun t : ℝ => T ≤ t ∧ comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta ≤ t) filter_upwards [eventually_ge_atTop T, eventually_ge_atTop (comparisonLowerSourceCoefficient A Cfirst Ctest Ccompare Cres 1 Q a ha delta)] with t ht hscale exact ⟨ht,hscale⟩ filter_upwards [happlicable delta hdelt hdhalf, actual_Q_approximation_eventual_margins (G+1) BQ H hd hdcompare hc ha N (by omega) delta,hlarge] with tau hactual hmarg hlargeTau rcases hmarg with ⟨ht2,_,_,heps1,hepsSmall,hepsScale,_,hepsQ1,hepsQSmall,hxxSmall⟩ have ht1 : 1 ≤ tau := by omega have htR : (1 : ℝ) ≤ tau := by exact_mod_cast ht1 refine ⟨ht1,?_⟩ intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ P hres p hp let g := perturbedMetric g0 eta.val have hSU : modelSquare ⊆ U := hf.squareSubset.trans hf.domainSubset have hOU : modelOpenSquare ⊆ U := modelOpenSquare_subset.trans hSU have hOV : modelOpenSquare ⊆ V := modelOpenSquare_subset.trans hSV have hgO : SmoothPositiveOn g modelOpenSquare := ⟨fun i j => (hg.1 i j).mono hOU,fun p hp => hg.2 p (hOU hp)⟩ have hgStarO : SmoothPositiveOn gStar modelOpenSquare := ⟨fun i j => (hgStar.1 i j).mono hOV,fun p hp => hgStar.2 p (hOV hp)⟩ have hzO := hh.smooth.mono hOU have hpoint := hactual.2.2.2 g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ have hb := hsourceT (tau : ℝ) hlargeTau.1 gStar g q0 modelOpenSquare hgStarO hactual.2.2.1 hgO modelOpenSquare_isOpen z P hzO (metricApproximationAccuracy tau) (qPulseFirstJetBudget a ha N delta (tau : ℝ)) heps1 hepsSmall hepsQ1 hepsQSmall hxxSmall p (hpoint p hp).1 (hres p hp) apply hb.trans apply comparisonSourceBudget_le_leading (E := 1) (Q := Q) ha N hA hCf hCc hCres htR · simpa only [one_mul] using hepsScale · exact le_rfl · exact hlargeTau.2 end SmoothLocal.Perturbation end end OAI