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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 | import OAI.Geometry.IsometricImmersion.Metrics.ModelCoercivity
namespace OAI
noncomputable section
open scoped ContDiff
namespace SmoothLocal.Model
open SmoothLocal.Geometry SmoothLocal.Weighted
def modelPrincipalRemainderBound (ell : ℕ) (R MG : ℝ) : ℝ :=
((ell : ℝ) + 1 / 2 + R / 2) * MG
def modelPrincipalRadius (g0 kappa rhoMax epsilon R MG : ℝ) (ell : ℕ) : ℝ :=
min (kappa / 8) (modelCoercivityMargin g0 kappa rhoMax epsilon ell /
(2 * (modelPrincipalRemainderBound ell R MG + 1)))
theorem modelPrincipalRemainderBound_nonneg (ell : ℕ) {R MG : ℝ}
(hR : 0 ≤ R) (hMG : 0 ≤ MG) : 0 ≤ modelPrincipalRemainderBound ell R MG := by
unfold modelPrincipalRemainderBound
positivity
theorem modelPrincipalRadius_pos (ell : ℕ) {g0 kappa rhoMax epsilon R MG : ℝ}
(hg0 : 0 < g0) (hkappa : 0 < kappa) (hrho : 0 < rhoMax) (heps : 0 < epsilon)
(hR : 0 ≤ R) (hMG : 0 ≤ MG) :
0 < modelPrincipalRadius g0 kappa rhoMax epsilon R MG ell := by
have hmu := modelCoercivityMargin_pos ell hg0 hkappa hrho heps
have hM := modelPrincipalRemainderBound_nonneg ell hR hMG
unfold modelPrincipalRadius
positivity
theorem modelPrincipal_derivative_remainder_bound
{G : Coord → ℝ} {p : Coord} (ell : ℕ) {epsilon R MG : ℝ}
(heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) (ht : |p 0| ≤ R)
(hGs : |coordPartial 1 G p| ≤ MG) (hGt : |coordPartial 0 G p| ≤ MG) :
|-((ell : ℝ) + 1 / 2) * coordPartial 1 G p +
epsilon * p 0 * coordPartial 0 G p / 2| ≤ modelPrincipalRemainderBound ell R MG := by
have hR : 0 ≤ R := (abs_nonneg _).trans ht
have hMG : 0 ≤ MG := (abs_nonneg _).trans hGs
have hbeta : 0 ≤ (ell : ℝ) + 1 / 2 := by positivity
have hfirst : |-((ell : ℝ) + 1 / 2) * coordPartial 1 G p| ≤
((ell : ℝ) + 1 / 2) * MG := by
rw [abs_mul, abs_neg, abs_of_nonneg hbeta]
exact mul_le_mul_of_nonneg_left hGs hbeta
have hprod : |epsilon * p 0 * coordPartial 0 G p| ≤ R * MG := by
rw [abs_mul, abs_mul, abs_of_nonneg heps]
have het : epsilon * |p 0| ≤ R :=
(mul_le_mul_of_nonneg_left ht heps).trans (by nlinarith)
exact mul_le_mul het hGt (abs_nonneg _) hR
have hsecond : |epsilon * p 0 * coordPartial 0 G p / 2| ≤ R * MG / 2 := by
rw [abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2)]
exact div_le_div_of_nonneg_right hprod (by norm_num)
have hsum := (abs_add_le _ _).trans (add_le_add hfirst hsecond)
unfold modelPrincipalRemainderBound
nlinarith only [hsum]
theorem modelPrincipal_radius_error
{g0 kappa rhoMax epsilon R MG K : ℝ} (ell : ℕ)
(hg0 : 0 < g0) (hkappa : 0 < kappa) (hrho : 0 < rhoMax) (heps : 0 < epsilon)
(hR : 0 ≤ R) (hMG : 0 ≤ MG)
(hK : |K| ≤ modelPrincipalRadius g0 kappa rhoMax epsilon R MG ell) :
|K| * modelPrincipalRemainderBound ell R MG ≤
modelCoercivityMargin g0 kappa rhoMax epsilon ell / 2 := by
let mu := modelCoercivityMargin g0 kappa rhoMax epsilon ell
let M := modelPrincipalRemainderBound ell R MG
have hmu : 0 < mu := modelCoercivityMargin_pos ell hg0 hkappa hrho heps
have hM : 0 ≤ M := modelPrincipalRemainderBound_nonneg ell hR hMG
have hden : 0 < 2 * (M + 1) := by positivity
have hK' : |K| ≤ mu / (2 * (M + 1)) := hK.trans (min_le_right _ _)
calc
_ ≤ mu / (2 * (M + 1)) * M := mul_le_mul_of_nonneg_right hK' hM
_ ≤ mu / (2 * (M + 1)) * (M + 1) :=
mul_le_mul_of_nonneg_left (by linarith) (div_pos hmu hden).le
_ = mu / 2 := by field_simp [show M + 1 ≠ 0 by positivity]
theorem model_full_principal_lower
{G K : Coord → ℝ} {p : Coord} {y kappa g0 rho rhoMax epsilon R MG Mq q : ℝ}
(ell : ℕ) (hGdiff : DifferentiableAt ℝ G p) (hKdiff : DifferentiableAt ℝ K p)
(hkappa : 0 < kappa) (hg0 : 0 < g0) (hG : g0 ≤ G p)
(hrho : 0 < rho) (hrhoMax : rho ≤ rhoMax) (heps : 0 < epsilon) (heps1 : epsilon ≤ 1)
(ht : |p 0| ≤ R) (hq : |q| ≤ Mq)
(hsmall : epsilon * R * Mq ≤ ((ell : ℝ) + 1 / 2) / rhoMax)
(hy : y ≤ 1 / 50)
(hGsi : |coordPartial 1 G p| ≤ MG) (hGti : |coordPartial 0 G p| ≤ MG)
(hKvalue : K p = modelCurvature kappa ![p 0, y])
(hKs : coordPartial 1 K p = coordPartial 1 (modelCurvature kappa) ![p 0, y] / rho)
(hKt : coordPartial 0 K p = 2 * kappa * p 0 -
q * coordPartial 1 (modelCurvature kappa) ![p 0, y])
(hKsmall : |K p| ≤ modelPrincipalRadius g0 kappa rhoMax epsilon R MG ell) :
modelCoercivityMargin g0 kappa rhoMax epsilon ell / 2 ≤
directedPrincipal (fun z => G z * K z) ell epsilon p := by
have hmax : 0 < rhoMax := hrho.trans_le hrhoMax
have hR : 0 ≤ R := (abs_nonneg _).trans ht
have hMG : 0 ≤ MG := (abs_nonneg _).trans hGsi
have hmodel : |modelCurvature kappa ![p 0, y]| ≤ kappa / 8 := by
rw [← hKvalue]
exact hKsmall.trans (min_le_left _ _)
have hgeo := model_geometric_principal_lower (p := ![p 0, y]) ell hkappa hg0 hG
hrho hrhoMax heps (by simpa using ht) hq hsmall (by simpa using hy) hmodel
simp only [Matrix.cons_val_zero] at hgeo
have hcoeff := modelPrincipal_derivative_remainder_bound ell heps.le heps1 ht hGsi hGti
have herr : |K p * (-((ell : ℝ) + 1 / 2) * coordPartial 1 G p +
epsilon * p 0 * coordPartial 0 G p / 2)| ≤
modelCoercivityMargin g0 kappa rhoMax epsilon ell / 2 := by
rw [abs_mul]
exact (mul_le_mul_of_nonneg_left hcoeff (abs_nonneg _)).trans
(modelPrincipal_radius_error ell hg0 hkappa hmax heps hR hMG hKsmall)
have heq : directedPrincipal (fun z => G z * K z) ell epsilon p =
G p * (-(((ell : ℝ) + 1 / 2) / rho + epsilon * p 0 * q / 2) *
coordPartial 1 (modelCurvature kappa) ![p 0, y] + epsilon * kappa * (p 0) ^ 2) +
K p * (-((ell : ℝ) + 1 / 2) * coordPartial 1 G p +
epsilon * p 0 * coordPartial 0 G p / 2) := by
unfold directedPrincipal
rw [HessianCalculus.coordPartial_mul_at hGdiff hKdiff 1,
HessianCalculus.coordPartial_mul_at hGdiff hKdiff 0, hKs, hKt]
field_simp [hrho.ne']
ring
rw [heq]
linarith [(abs_le.mp herr).1]
end SmoothLocal.Model
end
end OAI
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