import OAI.Geometry.IsometricImmersion.Taylor.ActualUniformTaylorRich import OAI.Geometry.IsometricImmersion.Comparison.ActualClassComparisonSourceBound import OAI.Geometry.IsometricImmersion.Comparison.ActualClassComparisonGradient 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_uniform_comparison_gradient {gStar : MetricField} {V : Set Coord} (hgStar : SmoothPositiveOn gStar V) (hV : IsOpen V) (hSV : modelSquare ⊆ V) {G d kappa q0 : 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) : ∃ r : ℝ, 0 < r ∧ r < 1/2 ∧ boundedClassWidth kappa M*r ≤ 1/20 ∧ heightQuotientJetBound G (M : ℝ) d (1/(M : ℝ))* (r+107*(boundedClassWidth kappa M*r)/100) ≤ 9/(100*boundedClassWidth kappa M) ∧ ∀ N : ℕ, 2 < N → ∃ Cres Csource Cgradient : ℝ, 0 ≤ Cres ∧ 0 < Csource ∧ 0 ≤ Cgradient ∧ ∀ delta : ℝ, 0 < delta → delta ≤ 1/2 → ∀ᶠ tau : ℕ in atTop, ∀ (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) → let zs := heightInShearCoordinates z q0 let gs := metricInShearCoordinates gStar q0 let P := taylorApproximation gs (-delta/(tau : ℝ)) (heightCauchyValue zs (-delta/(tau : ℝ))) (heightCauchyVelocity zs (-delta/(tau : ℝ))) N ContDiffOn ℝ ∞ P (spatialStrip (Ioo (-(boundedClassWidth kappa M*r)) (boundedClassWidth kappa M*r))) ∧ (∀ x ∈ Ioo (-(boundedClassWidth kappa M*r)) (boundedClassWidth kappa M*r), P ![x,-delta/(tau : ℝ)] = zs ![x,-delta/(tau : ℝ)] ∧ coordPartial 1 P ![x,-delta/(tau : ℝ)] = coordPartial 1 zs ![x,-delta/(tau : ℝ)]) ∧ (∀ x, ∃ Q : Polynomial ℝ, Q.natDegree ≤ N+1 ∧ ∀ t, Q.eval t = P ![x,t]) ∧ (∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ t ∈ Icc (-delta/(tau : ℝ)) (delta/(tau : ℝ)), (boundedClassSpeed kappa M)^2/(8*(M : ℝ)) ≤ |covHessian gs P ![x,t] 0 0| ∧ |qResidual gs P ![x,t]| ≤ Cres/(tau : ℝ)^N) ∧ ContinuousOn (qResidual gs P) (pulseStrip (boundedClassWidth kappa M*r/16) delta (tau : ℝ)) ∧ |pulseWeightedMoment (boundedClassWidth kappa M*r/16) delta (tau : ℝ) (qResidual gs P)| ≤ (2*Cres*(∫ x : ℝ, axisBump (boundedClassWidth kappa M*r/16) x))*delta/(tau : ℝ)^(N+1) ∧ (∀ 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) ∧ (∀ theta ∈ Icc (-(delta/(tau : ℝ))) (delta/(tau : ℝ)), ∀ i : Fin 2, Real.sqrt (∫ x in Icc (-(boundedClassWidth kappa M*r/16)) (boundedClassWidth kappa M*r/16), (coordPartial i (comparisonDifference P zs) (boxPoint x theta))^2) ≤ Cgradient*delta*(tau : ℝ)/(tau : ℝ)^N) := by obtain ⟨r,hr,hrhalf,hLr,hrsmall,hTaylor⟩ := exists_actual_uniform_taylor_comparison_and_moment hgStar hV hSV M hG hd hkappa hM hq0 refine ⟨r,hr,hrhalf,hLr,hrsmall,?_⟩ intro N hN obtain ⟨Cres,hCres,hTaylorN⟩ := hTaylor N obtain ⟨Csource,hCsource,hsource⟩ := exists_actual_class_comparison_source_bound_at_radius hgStar hV hSV M hG hd hgStarB hdStar hkappa hM hq0 hr hrhalf hLr hrsmall N hN Cres hCres obtain ⟨CE,hCE,hgradient⟩ := exists_actual_class_comparison_gradient_from_source hgStar hV hSV M hG hd hgStarB hdStar hkappa hM hq0 hr hrhalf hLr hrsmall N hN refine ⟨Cres,Csource,CE*Csource,hCres,hCsource,mul_nonneg hCE hCsource.le,?_⟩ intro delta hdelt hdhalf filter_upwards [hTaylorN delta hdelt hdhalf,hsource delta hdelt hdhalf,hgradient delta hdelt hdhalf] with tau htaylor hsourceTau hgradientTau intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ let L := boundedClassWidth kappa M let zs := heightInShearCoordinates z q0 let gs := metricInShearCoordinates gStar q0 let P := taylorApproximation gs (-delta/(tau : ℝ)) (heightCauchyValue zs (-delta/(tau : ℝ))) (heightCauchyVelocity zs (-delta/(tau : ℝ))) N have ht := htaylor g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ have hres (p : Coord) (hp : p ∈ pulseStrip (L*r/2) delta (tau : ℝ)) : |qResidual gs P p| ≤ Cres/(tau : ℝ)^N := by have he : ![p 0,p 1] = p := by ext i; fin_cases i <;> rfl have htime : p 1 ∈ Icc (-delta/(tau : ℝ)) (delta/(tau : ℝ)) := by simpa only [neg_div] using hp.2 simpa only [he] using (ht.2.2.2.1 (p 0) (abs_le.mpr hp.1) (p 1) htime).2 have hs := hsourceTau.2 g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ P hres refine ⟨ht.1,ht.2.1,ht.2.2.1,ht.2.2.2.1,ht.2.2.2.2.1,ht.2.2.2.2.2,hs,?_⟩ have htpos : (0 : ℝ) < tau := zero_lt_one.trans_le (by exact_mod_cast hsourceTau.1) have hF : 0 ≤ Csource*(tau : ℝ)^2/(tau : ℝ)^N := by positivity intro theta htheta i have hgrad := hgradientTau g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ (Csource*(tau : ℝ)^2/(tau : ℝ)^N) hF hs theta htheta i apply hgrad.trans_eq field_simp [htpos.ne'] end SmoothLocal.Perturbation end end OAI