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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 | import OAI.Geometry.IsometricImmersion.Metrics.SmoothMetricRestriction
import OAI.Geometry.IsometricImmersion.Coordinates.ActualReferenceSegment
import OAI.Geometry.IsometricImmersion.Comparison.ActualComparisonGradient
import OAI.Geometry.IsometricImmersion.Comparison.ComparisonPulseGeometry
import OAI.Geometry.IsometricImmersion.Taylor.TaylorInitialVelocity
namespace OAI
noncomputable section
open Set Filter Function MeasureTheory
open scoped ContDiff Topology Matrix
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_gradient_from_source
{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) :
∃ Cgradient : ℝ, 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
∀ F : ℝ, 0 ≤ F →
(∀ p ∈ pulseStrip (boundedClassWidth kappa M*r/2) delta (tau : ℝ),
|actualComparisonSource gStar (perturbedMetric g0 eta.val) z q0 P p| ≤ F) →
∀ 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*F*delta/(tau : ℝ) := by
let L := boundedClassWidth kappa M
let Z := (2*(1+|q0|))^3*(M : ℝ)
let nu := (boundedClassSpeed kappa M)^2/(2*(M : ℝ))
let gs := metricInShearCoordinates gStar q0
let Vs := inverseShearCoordinates q0 ⁻¹' V
have hL : 0 < L := boundedClassWidth_pos kappa M
have hLrpos : 0 < L*r := mul_pos hL hr
have hnu : 0 < nu := boundedClassShearHxxFloor_pos hkappa hM
have hminv : 0 < 1/(M : ℝ) := one_div_pos.mpr (Nat.cast_pos.mpr hM)
obtain ⟨CP,A,_,hA,hsegment⟩ := exists_actual_reference_segment_at_radius
hgStar hV hSV M hG hd hgStarB hdStar hkappa hM hq0 hr hrhalf hLr hrsmall N hN
let s0 := ((nu*(1/(M : ℝ)))/2)/A^2
have hs0 : 0 < s0 := div_pos (half_pos (mul_pos hnu hminv))
(sq_pos_of_pos (zero_lt_one.trans_le hA))
have hgs : SmoothPositiveOn gs Vs := metricInShearCoordinates_smoothPositive hgStar q0
have hVs : IsOpen Vs := shearedModelDomain_isOpen hV q0
obtain ⟨CE,hCE,hgradient⟩ := exists_actual_comparison_gradient hgs hVs
(shearedModelSquare_isCompact q0) (shearedModelSquare_subset_domain hSV q0)
1 CP Z (by norm_num) (half_pos hnu) hs0
let speed := comparisonEnergySpeed CE
let rate := comparisonEnergyRate CE CP Z s0
let Cgradient := 2*Real.exp 1*Real.sqrt (2*(1+1/s0))*Real.sqrt (2*(L*r/4))
have hs : 0 ≤ speed := add_nonneg (zero_le_one.trans hCE) (Real.sqrt_nonneg _)
have hCg : 0 ≤ Cgradient := by dsimp [Cgradient]; positivity
refine ⟨Cgradient,hCg,?_⟩
intro delta hdelt hdhalf
have hwidth : ∀ᶠ tau : ℕ in atTop, 1 ≤ (tau : ℝ) ∧ delta/(2*(tau : ℝ)) ≤ r/4 :=
(tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually
(pulse_width_eventually hr delta)
have hlarge : ∀ᶠ tau : ℕ in atTop, rate*(2*delta) ≤ (tau : ℝ) :=
(tendsto_natCast_atTop_atTop : Tendsto (fun tau : ℕ => (tau : ℝ)) atTop atTop).eventually
(eventually_ge_atTop _)
have hpulse : 0 < L*r/16 := div_pos hLrpos (by norm_num : (0 : ℝ) < 16)
filter_upwards [hsegment delta hdelt hdhalf,hwidth,hlarge,
comparison_pulse_geometry_eventually hpulse hs delta hdelt.le]
with tau hsegTau hw hrateTau htravel
intro g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ
let g := perturbedMetric g0 eta.val
let gt := metricInShearCoordinates g q0
let zs := heightInShearCoordinates z q0
let a := -delta/(tau : ℝ)
let b := delta/(tau : ℝ)
let I := Ioo (-(L*r)) (L*r)
let P := taylorApproximation gs a (heightCauchyValue zs a) (heightCauchyVelocity zs a) N
let T := closedRectangle (-(L*r/2)) (L*r/2) a b
have hTP : T = pulseStrip (L*r/2) delta (tau : ℝ) := by
simp only [T,pulseStrip,a,b,neg_div]
let Us := inverseShearCoordinates q0 ⁻¹' U
let D := (Us ∩ Vs) ∩ spatialStrip I
have hUs : IsOpen Us := shearedModelDomain_isOpen hU q0
have hD : IsOpen D := (hUs.inter hVs).inter (spatialStrip_isOpen isOpen_Ioo)
have hSU : modelSquare ⊆ U := hf.squareSubset.trans hf.domainSubset
have htpos : (0 : ℝ) < tau := zero_lt_one.trans_le hw.1
have hbnonneg : 0 ≤ b := div_nonneg hdelt.le htpos.le
have hab : a ≤ b := by dsimp only [a,b]; simp only [neg_div]; linarith
have hbr : b ≤ r := by
have he : delta/(tau : ℝ)=2*(delta/(2*(tau : ℝ))) := by ring
dsimp only [b]
rw [he]
linarith [hw.2]
have hcoordinates (p : Coord) (hp : p ∈ T) : |p 0| ≤ L*r ∧ |p 1| ≤ r := by
have hx : |p 0| ≤ L*r/2 := abs_le.mpr hp.1
have ht : |p 1| ≤ b := abs_le.mpr
⟨by simpa only [a,b,neg_div] using hp.2.1,hp.2.2⟩
exact ⟨hx.trans (by linarith),ht.trans hbr⟩
have hgeom (p : Coord) (hp : p ∈ T) := actual_central_sheared_solution_margins
eta hclass hG hd hkappa hr.le hLr hq0 hrsmall hgB hdet hcenter
(hcoordinates p hp).1 (hcoordinates p hp).2
have hpS (p : Coord) (hp : p ∈ T) : inverseShearCoordinates q0 p ∈ modelSquare := (hgeom p hp).1
have hbox : T ⊆ D := by
intro p hp
exact ⟨⟨hSU (hpS p hp),hSV (hpS p hp)⟩,
⟨by linarith [hp.1.1],by linarith [hp.1.2]⟩⟩
have hpoint (p : Coord) (hp : p ∈ T) : ‖p‖ ≤ 1 := by
apply (pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 1)).mpr
intro i
fin_cases i
· change |p 0| ≤ 1
exact (hcoordinates p hp).1.trans (hLr.trans (by norm_num))
· change |p 1| ≤ 1
exact (hcoordinates p hp).2.trans (by linarith)
have hgt : SmoothPositiveOn gt D :=
(metricInShearCoordinates_smoothPositive hg q0).mono (fun p hp => hp.1.1)
have hzsUs : ContDiffOn ℝ ∞ zs Us := heightInShearCoordinates_contDiffOn hh.smooth q0
have hzsD : ContDiffOn ℝ ∞ zs D := hzsUs.mono (fun p hp => hp.1.1)
obtain ⟨hP,hPB,hsegActual⟩ := hsegTau g0 eta U W z Y hg hU hh hf hclass hgB hdet hcenter happ
have hPD : ContDiffOn ℝ ∞ P D := hP.mono inter_subset_right
have hzB : CoordinateBound zs T 3 Z := bounded_class_sheared_C3_on_actual_region hclass q0 hpS
have hPB' : CoordinateBound P T 3 CP := by rw [hTP]; exact hPB
have hCauchy := actual_taylor_Cauchy_of_smooth gs zs a N isOpen_Ioo hP
have hpointEq (x t : ℝ) : boxPoint x t=(![x,t] : Coord) := by
ext i
fin_cases i
· change x*1+t*0=x
ring
· change x*0+t*1=t
ring
have hvalue (x : ℝ) (hx : x ∈ Ioo (-(L*r/2)) (L*r/2)) :
P (boxPoint x a)=zs (boxPoint x a) := by
rw [hpointEq]
exact (hCauchy x ⟨by linarith [hx.1],by linarith [hx.2]⟩).1
have hvelocity (x : ℝ) (hx : x ∈ Ioo (-(L*r/2)) (L*r/2)) :
coordPartial 1 P (boxPoint x a)=coordPartial 1 zs (boxPoint x a) := by
rw [hpointEq]
exact (hCauchy x ⟨by linarith [hx.1],by linarith [hx.2]⟩).2
have hmargin (p : Coord) (hp : p ∈ T) (sigma : ℝ) (hsigma : sigma ∈ Icc (0 : ℝ) 1) :
nu/2 ≤ |stateQDenominator gs (qHeightJetSegment P zs sigma p)| ∧
s0 ≤ qFirstCoefficient gs 5 (qHeightJetSegment P zs sigma p) := by
have hm := hsegActual p (by simpa only [hTP] using hp) sigma hsigma
have he : nu/2=(boundedClassSpeed kappa M)^2/(4*(M : ℝ)) := by dsimp [nu]; ring
exact ⟨by simpa only [he] using hm.1,hm.2.2.2.1⟩
have htravel' : speed*(b-a) ≤ L*r/4-L*r/16 := by
have he : b-a=2*delta/(tau : ℝ) := by dsimp [a,b]; ring
rw [he]
calc
speed*(2*delta/(tau : ℝ)) ≤ 3*(L*r/16) := htravel.2.1
_ = L*r/4-L*r/16 := by ring
dsimp only
intro F hF hsource theta htheta i
have hsourceMetric (p : Coord) (hp : p ∈ T) :
|metricComparisonSource gt gs P zs p| ≤ F := by
change |actualComparisonSource gStar g z q0 P p| ≤ F
exact hsource p (by simpa only [hTP] using hp)
have hgrad := hgradient gt P zs D (L*r/2) (L*r/4) (L*r/16) a b F
hD (fun p hp => hp.1.2) hgt hPD hzsD (by positivity) (by linarith) hab
hbox (fun p hp => hpS p hp) hpoint hPB' hzB
(fun p hp => (hgeom p hp).2.2.2.2.2.1)
(fun p hp => abs_pos.mp (hnu.trans_le (hgeom p hp).2.1))
hmargin hvalue hvelocity (by positivity) hF htravel'
hsourceMetric theta
(by simpa only [a,b,neg_div] using htheta) i
have he : b-a=2*delta/(tau : ℝ) := by dsimp [a,b]; ring
rw [he] at hgrad
have hexp := comparison_exponential_le_fixed htpos hrateTau
have hbound := mul_le_mul_of_nonneg_right hexp hF
have hbound := mul_le_mul_of_nonneg_right hbound (Real.sqrt_nonneg (2*(L*r/4)))
have hduration : 0 ≤ 2*delta/(tau : ℝ) :=
div_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hdelt.le) htpos.le
have hbound := mul_le_mul_of_nonneg_right hbound hduration
have hbound := mul_le_mul_of_nonneg_left hbound (Real.sqrt_nonneg (2*(1+1/s0)))
apply hgrad.trans
convert hbound using 1
dsimp [Cgradient]
ring
end SmoothLocal.Perturbation
end
end OAI
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