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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 | import OAI.Geometry.IsometricImmersion.Calculus.BoundedClassInitialJet
import OAI.Geometry.IsometricImmersion.Pulses.PulseEdgeReferenceDistance
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology Matrix Matrix.Norms.Elementwise
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.HighEquation SmoothLocal.Perturbation
open SmoothLocal.Flow SmoothLocal.Taylor
theorem exists_bounded_patch_reference_initial_floor
(G kappa d q0 : ℝ) (M : ℕ) (hG : 0 ≤ G)
(hkappa : 0 < kappa) (hd : 0 < d) (hM : 0 < M) (hq0 : |q0| ≤ 1 / 20) :
∃ tau0 : ℕ, 1 ≤ tau0 ∧
∀ (g0 gStar : MetricField) (V : Set Coord) (eta : metricPatchSet g0 kappa)
(z : Coord → ℝ) (r : ℝ) (N : ℕ) (delta : ℝ) (tau : ℕ),
SmoothPositiveOn gStar V → IsOpen V → SmoothLocal.Flow.modelSquare ⊆ V →
BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z →
0 < r → 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) →
(∀ i j k, k ≤ 4 → ∀ p ∈ SmoothLocal.Flow.modelSquare,
‖iteratedFDeriv ℝ k (fun a => perturbedMetric g0 eta.val a i j) p‖ ≤ G) →
(∀ p ∈ SmoothLocal.Flow.modelSquare, d ≤ |(perturbedMetric g0 eta.val p).det|) →
|hessianQuotient (perturbedMetric g0 eta.val) z 0 - q0| ≤
1 / (100 * boundedClassWidth kappa M) →
0 < delta → tau0 ≤ tau → delta / (tau : ℝ) ≤ r →
(∀ i j k, k ≤ tau → ∀ p ∈ SmoothLocal.Flow.modelSquare,
‖iteratedFDeriv ℝ k
(fun a => perturbedMetric g0 eta.val a i j -
testMetric gStar q0 (boundedClassWidth kappa M * r / 16) N delta (tau : ℝ) a i j) p‖ ≤
metricApproximationAccuracy tau) →
∀ x ∈ Ioo (-(boundedClassWidth kappa M * r)) (boundedClassWidth kappa M * r),
(boundedClassSpeed kappa M)^2 / (4 * (M : ℝ)) ≤
|covHessian (metricInShearCoordinates gStar q0)
(linearCauchy (-delta / (tau : ℝ))
(heightCauchyValue (heightInShearCoordinates z q0) (-delta / (tau : ℝ)))
(heightCauchyVelocity (heightInShearCoordinates z q0) (-delta / (tau : ℝ))))
![x, -delta / (tau : ℝ)] 0 0| := by
let c := (boundedClassSpeed kappa M)^2 / (2 * (M : ℝ))
have hc : 0 < c := boundedClassShearHxxFloor_pos hkappa hM
obtain ⟨epsilon, he, htol⟩ := exists_reference_linearCauchy_XX_tolerance
(8 * G) (boundedClassShearedStateBudget q0 M) hd hc
obtain ⟨tau0, htau0, herror⟩ := exists_first_error_threshold he
refine ⟨tau0, htau0, ?_⟩
intro g0 gStar V eta z r N delta tau hgStar hV hSV hclass hr hLr hrsmall
hgB hdet hcenter hdelta htau hwidth happrox x hx
let g := perturbedMetric g0 eta.val
let L := boundedClassWidth kappa M
let a := -delta / (tau : ℝ)
have hclass0 := hclass
obtain ⟨_, hadm, _, _, _⟩ := hclass
obtain ⟨U, hU, hSU, hg, hz, _, _, _, _⟩ := hadm
have hL : 0 < L := boundedClassWidth_pos kappa M
have hL1 : 1 ≤ L := (by norm_num : (1 : ℝ) ≤ 100).trans (boundedClassWidth_ge_hundred kappa M)
have hrsmall' : r ≤ 1 / 20 := by
have hh : r ≤ L * r := by nlinarith
exact hh.trans hLr
have htauN : 1 ≤ tau := htau0.trans htau
have htauR : (0 : ℝ) < (tau : ℝ) := zero_lt_one.trans_le (by exact_mod_cast htauN)
have habs : |a| ≤ r := by
dsimp [a]
rw [abs_div, abs_neg, abs_of_pos hdelta, abs_of_pos htauR]
exact hwidth
have hcutCentral (y : ℝ) (hy : y ∈ Ioo (-(L * r)) (L * r)) :
inverseShearCoordinates q0 ![y, a] ∈ centralBox :=
inverseShear_slab_mem_centralBox hL1 hr.le hLr hq0 (abs_le.mpr ⟨hy.1.le, hy.2.le⟩) habs
have hcutS (y : ℝ) (hy : y ∈ Ioo (-(L * r)) (L * r)) :
inverseShearCoordinates q0 ![y, a] ∈ SmoothLocal.Flow.modelSquare :=
centralBox_subset_modelSquare (hcutCentral y hy)
have hpS := hcutS x hx
have hpU := hSU hpS
have hpx : |x| ≤ L * r := abs_le.mpr ⟨hx.1.le, hx.2.le⟩
have hsmall := bounded_class_slab_quotient_small_of_cap_radius hclass0 hG hd hr.le hq0 hrsmall
hgB hdet hcenter hpS hpx habs
have hK : gaussianCurvature g (inverseShearCoordinates q0 ![x, a]) ≤ -kappa / 2 :=
(eta.property.2 _ (hcutCentral x hx)).le
obtain ⟨_, hxx, _, _⟩ := bounded_class_shear_estimates hclass0 hkappa hpS hK hsmall
have hq1 : |q0| ≤ 1 := hq0.trans (by norm_num)
have hfirst := sheared_first_input_norm_le_of_original_jets hg hU hpU hG hq1
(fun i j k hk => hgB i j k (by omega) _ hpS)
have hstate := bounded_class_sheared_solution_state_bound hclass0 q0 hpS
(show |(![x,a] : Coord) 0| ≤ 1 ∧ |(![x,a] : Coord) 1| ≤ 1 from
⟨by change |x| ≤ 1; exact hpx.trans (hLr.trans (by norm_num)),
by change |a| ≤ 1; exact habs.trans (hrsmall'.trans (by norm_num))⟩)
have hdet0 : d ≤ (g (inverseShearCoordinates q0 ![x,a])).det := by
have hh := hdet _ hpS
rw [abs_of_pos (hg.2 _ hpU).det_pos] at hh
exact hh
have hdetS : d ≤ (metricInShearCoordinates g q0 ![x,a]).det := by
rw [metricDet_in_shear_coordinates]
exact hdet0
have hdist := actual_pulse_edge_reference_first_distance hg hgStar hU hV hq1
(div_pos (mul_pos hL hr) (by norm_num)) hdelta htauN hpU (hSV hpS)
(fun i j k hk => happrox i j k hk _ hpS)
have hUS := hU.preimage (inverseShearCoordinates_contDiff q0).continuous
have hzS := heightInShearCoordinates_contDiffOn hz q0
have hresult := htol (metricInShearCoordinates g q0) (metricInShearCoordinates gStar q0)
(heightInShearCoordinates z q0) (inverseShearCoordinates q0 ⁻¹' U) hzS hUS
(-(L * r)) (L * r) a (fun y hy => hSU (hcutS y hy)) x hx hfirst hstate hdetS hxx
(hdist.trans (herror tau htau))
have heq : c / 2 = (boundedClassSpeed kappa M)^2 / (4 * (M : ℝ)) := by
dsimp [c]
ring
rw [heq] at hresult
exact hresult
end SmoothLocal.Pulse
end
end OAI
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