| 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 |
|
|
