openAI-math-dataset / lean /OAI /Geometry /IsometricImmersion /Comparison /ActualUniformComparisonGradient.lean
| 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 | |