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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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 | import OAI.Geometry.IsometricImmersion.Comparison.ActualClassComparisonMoment
import OAI.Geometry.IsometricImmersion.Caps.ActualClassTwoSidedCauchy
import OAI.Geometry.IsometricImmersion.Taylor.ActualTaylorFiniteJets
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 SmoothLocal.Model
theorem exists_original_class_comparison_with_low_data
{g0 gStar : MetricField} {V : Set Coord}
(hg0 : SmoothPositiveOn g0 V) (hgStar : SmoothPositiveOn gStar V)
(hV : IsOpen V) (hSV : modelSquare ⊆ V)
{kappa q0 : ℝ} (M : ℕ) (hkappa : 0 < kappa) (hM : 0 < M)
(hq0 : |q0| ≤ 1/20)
(hbackground : ∀ p ∈ V, gaussianCurvature g0 p = modelCurvature kappa p) :
∃ G d r : ℝ, 0 < G ∧ 0 < d ∧ 0 < r ∧ r < 1/2 ∧
(∀ i j : Fin 2, ∀ k ≤ 8, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ G) ∧
(∀ p ∈ modelSquare, d ≤ (gStar p).det) ∧
(∀ a : ℝ, 0 < a → ∀ N : ℕ, 10 < N → ∀ delta : ℝ,
∀ᶠ tau : ℕ in atTop,
∀ (gTau : MetricField) (U : Set Coord), SmoothPositiveOn gTau U → IsOpen U → modelSquare ⊆ U →
(∀ i j : Fin 2, ∀ k ≤ tau, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => gTau q i j-
testMetric gStar q0 a N delta (tau : ℝ) q i j) p‖ ≤ metricApproximationAccuracy tau) →
(∀ i j : Fin 2, ∀ k ≤ 8, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => gTau q i j) p‖ ≤ G) ∧
(∀ p ∈ modelSquare, d ≤ |(gTau p).det|)) ∧
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) ∧
(∀ K : ℕ, ∃ B : ℝ, 0 ≤ B ∧ ∀ N : ℕ, 10 < N → ∀ delta : ℝ, 0 < delta →
∀ᶠ tau : ℕ in atTop,
∀ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
|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) →
∀ x : ℝ, |x| ≤ boundedClassWidth kappa M*r/2 → ∀ n ≤ K,
(‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0)
(-delta/(tau : ℝ))) x‖ ≤ B ∧
‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0)
(-delta/(tau : ℝ))) x‖ ≤ B) ∧
(‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0)
(delta/(tau : ℝ))) x‖ ≤ B ∧
‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0)
(delta/(tau : ℝ))) x‖ ≤ B)) ∧
∀ N : ℕ, 10 < N → ∃ C Csource Cgradient c3 : ℝ,
0 ≤ C ∧ 0 < Csource ∧ 0 ≤ Cgradient ∧ 0 < c3 ∧
(∀ K : ℕ, ∃ CP : ℝ, 0 ≤ CP ∧
∀ delta : ℝ, 0 < delta → delta ≤ 1/2 → ∀ᶠ tau : ℕ in atTop,
∀ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
|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))) ∧
(∀ j ≤ K, ∀ p : Coord, |p 0| ≤ boundedClassWidth kappa M*r/2 →
|p 1| ≤ delta/(tau : ℝ) → ‖iteratedFDeriv ℝ j P p‖ ≤ CP) ∧
CoordinateBound P (pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ)) K CP) ∧
∀ delta : ℝ, 0 < delta → delta ≤ 1/2 → ∀ᶠ tau : ℕ in atTop,
∀ (eta : metricPatchSet g0 kappa) (z : Coord → ℝ),
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
|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, ∃ poly : Polynomial ℝ, poly.natDegree ≤ N+1 ∧ ∀ t, poly.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]| ≤ C/(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*C*(∫ 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) ∧
ContinuousOn (actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P)
(pulseStrip (boundedClassWidth kappa M*r/16) delta (tau : ℝ)) ∧
c3*delta*(tau : ℝ)/(tau : ℝ)^N ≤
|pulseWeightedMoment (boundedClassWidth kappa M*r/16) delta (tau : ℝ)
(actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P)| := by
obtain ⟨G,d,hG,hd,hGstar,hdstar,hlow,hcuts⟩ :=
exists_actual_class_two_sided_cauchy_bounds_at_every_radius
hg0 hgStar hV hSV M hkappa hM hq0 hbackground
have hGstar2 : ∀ i j : Fin 2, ∀ k ≤ 2, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => gStar q i j) p‖ ≤ G :=
fun i j k hk p hp => hGstar i j k (by omega) p hp
obtain ⟨r,hr,hrhalf,hLr,hrsmall,hcomparison⟩ :=
exists_actual_class_comparison_moment hgStar hV hSV M hG.le hd hGstar2 hdstar hkappa hM hq0
refine ⟨G,d,r,hG,hd,hr,hrhalf,hGstar,hdstar,hlow,hLr,hrsmall,
hcuts r hr hrhalf hLr hrsmall,?_⟩
intro N hN
obtain ⟨C,Csource,Cgradient,c3,hC,hCs,hCg,hc3,hcomparisonN⟩ := hcomparison N (by omega)
refine ⟨C,Csource,Cgradient,c3,hC,hCs,hCg,hc3,?_,?_⟩
· intro K
obtain ⟨CP,hCP,hPjets⟩ := exists_actual_taylor_finite_jets_at_radius
hgStar hV hSV M hG.le hd hkappa hM hq0 hr hrhalf hLr hrsmall N K
refine ⟨CP,hCP,?_⟩
intro delta hdelt hdhalf
have ha : 0 < boundedClassWidth kappa M*r/16 :=
div_pos (mul_pos (boundedClassWidth_pos kappa M) hr) (by norm_num)
filter_upwards [hPjets delta hdelt hdhalf,
hlow (boundedClassWidth kappa M*r/16) ha N hN delta] with tau hPjetsTau hlowTau
intro eta z hclass hcenter happ
obtain ⟨U,hU,hSU,hUV,_,hgU,hh⟩ := bounded_class_actual_cap_data hg0 hV hSV eta hclass
obtain ⟨hg8,hdet⟩ := hlowTau (perturbedMetric g0 eta.val) U hgU hU hSU happ
have hg4 : ∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j) p‖ ≤ G :=
fun i j k hk p hp => hg8 i j k (by omega) p hp
have hmodel : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p :=
fun p hp => hbackground p (hUV hp)
have hc : 0 < 1/(M : ℝ) := one_div_pos.mpr (Nat.cast_pos.mpr hM)
obtain ⟨W,Y,hf⟩ := exists_capInductionFlow hgU hU hSU hG.le (Nat.cast_nonneg M)
hd hc hc hg8 hdet hh hkappa hmodel (perturbationTensor_tsupport_subset eta.val) eta.property.2
exact hPjetsTau g0 eta U W z Y hgU hU hh hf hclass hg4 hdet hcenter happ
intro delta hdelt hdhalf
have ha : 0 < boundedClassWidth kappa M*r/16 := by
exact div_pos (mul_pos (boundedClassWidth_pos kappa M) hr) (by norm_num)
filter_upwards [hcomparisonN delta hdelt hdhalf,
hlow (boundedClassWidth kappa M*r/16) ha N hN delta] with tau hcTau hlowTau
intro eta z hclass hcenter happ
obtain ⟨U,hU,hSU,hUV,_,hgU,hh⟩ := bounded_class_actual_cap_data hg0 hV hSV eta hclass
obtain ⟨hg8,hdet⟩ := hlowTau (perturbedMetric g0 eta.val) U hgU hU hSU happ
have hg4 : ∀ i j : Fin 2, ∀ k ≤ 4, ∀ p ∈ modelSquare,
‖iteratedFDeriv ℝ k (fun q => perturbedMetric g0 eta.val q i j) p‖ ≤ G :=
fun i j k hk p hp => hg8 i j k (by omega) p hp
have hmodel : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p :=
fun p hp => hbackground p (hUV hp)
have hc : 0 < 1/(M : ℝ) := one_div_pos.mpr (Nat.cast_pos.mpr hM)
obtain ⟨W,Y,hf⟩ := exists_capInductionFlow hgU hU hSU hG.le (Nat.cast_nonneg M)
hd hc hc hg8 hdet hh hkappa hmodel (perturbationTensor_tsupport_subset eta.val) eta.property.2
exact hcTau g0 eta U W z Y hgU hU hh hf hclass hg4 hdet hcenter happ
end SmoothLocal.Perturbation
end
end OAI
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