| import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientRegularity |
| import OAI.Geometry.IsometricImmersion.Energy.DirectedCoercivity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff |
|
|
| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry |
|
|
| def directedPrincipalBound (ell : ℕ) (epsilon R MAs MAt : ℝ) : ℝ := |
| ((ell : ℝ) + 1 / 2) * MAs + epsilon * R * MAt / 2 |
|
|
| theorem directedPrincipal_abs_bound {A : Coord → ℝ} {p : Coord} |
| (ell : ℕ) {epsilon R MAs MAt : ℝ} (heps : 0 ≤ epsilon) |
| (ht : |p 0| ≤ R) (hAs : |coordPartial 1 A p| ≤ MAs) |
| (hAt : |coordPartial 0 A p| ≤ MAt) : |
| |directedPrincipal A ell epsilon p| ≤ directedPrincipalBound ell epsilon R MAs MAt := by |
| have hR : 0 ≤ R := (abs_nonneg _).trans ht |
| have hMAt : 0 ≤ MAt := (abs_nonneg _).trans hAt |
| have hbeta : 0 ≤ (ell : ℝ) + 1 / 2 := by positivity |
| have hfirst : |-((ell : ℝ) + 1 / 2) * coordPartial 1 A p| ≤ ((ell : ℝ) + 1 / 2) * MAs := by |
| rw [abs_mul, abs_neg, abs_of_nonneg hbeta] |
| exact mul_le_mul_of_nonneg_left hAs hbeta |
| have hsecond : |epsilon * p 0 * coordPartial 0 A p / 2| ≤ epsilon * R * MAt / 2 := by |
| rw [abs_div, abs_mul, abs_mul, abs_of_nonneg heps, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| exact div_le_div_of_nonneg_right |
| (mul_le_mul (mul_le_mul_of_nonneg_left ht heps) hAt (abs_nonneg _) (mul_nonneg heps hR)) |
| (by norm_num) |
| exact (abs_add_le _ _).trans (add_le_add hfirst hsecond) |
|
|
| theorem primitive_directed_farnegative_coercivity |
| {B : Coord → ℝ} {R a b : ℝ} {p : Coord} |
| (hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) (hR : 0 < R) |
| (hp : p ∈ coordinateRectangle R a b) (A C r : Coord → ℝ) |
| (hA : DifferentiableAt ℝ A p) (edge lambda epsilon : ℝ) (ell : ℕ) |
| (hedge : p 1 < edge) (hlambda : 0 < lambda) |
| (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) |
| (hC : C p = ((ell : ℝ) + 1) * coordPartial 1 A p + A p * r p) |
| (MI MB Mr MA MAs MAt mu alpha : ℝ) (hmu : 0 < mu) |
| (hIs : |coordPartial 1 (coordinatePrimitive B) p| ≤ MI) |
| (hBp : |B p| ≤ MB) (hr : |r p| ≤ Mr) |
| (hAp : |A p| ≤ MA) (hAs : |coordPartial 1 A p| ≤ MAs) |
| (hTbudget : 2 * (MI + epsilon + epsilon * R * MB) ≤ lambda) |
| (hEbudget : (MI + 1 + R * MB) / 2 + Mr ≤ lambda / 8) |
| (hsmall : 4 * (R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1)) ^ 2 * |
| epsilon ^ 2 * MA ≤ 1 / 8) |
| (hloss : (4 * (R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1)) ^ 2 * |
| epsilon ^ 2) / lambda ≤ mu / 4) |
| (hAt : |coordPartial 0 A p| ≤ MAt) |
| (halpha : 0 < alpha) (hAfar : A p ≤ -alpha) |
| (hdominance : directedPrincipalBound ell epsilon R MAs MAt + mu ≤ lambda * alpha / 4) |
| (x y : ℝ) : |
| min (lambda / 8) (mu / 2) * directedWeight edge lambda (coordinatePrimitive B) p * |
| (x ^ 2 + y ^ 2) ≤ |
| multiplierT B (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * x ^ 2 + |
| multiplierS A C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * y ^ 2 + |
| multiplierJ A B C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p * x * y := by |
| let W := directedWeight edge lambda (coordinatePrimitive B) p |
| let H := weightH0 edge lambda p |
| let Cj := R * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1) |
| let L := 4 * Cj ^ 2 * epsilon ^ 2 |
| let E := (MI + 1 + R * MB) / 2 + Mr |
| have hMI : 0 ≤ MI := (abs_nonneg _).trans hIs |
| have hMB : 0 ≤ MB := (abs_nonneg _).trans hBp |
| have hMr : 0 ≤ Mr := (abs_nonneg _).trans hr |
| have hMA : 0 ≤ MA := (abs_nonneg _).trans hAp |
| have hMAs : 0 ≤ MAs := (abs_nonneg _).trans hAs |
| have hCj : 0 ≤ Cj := by dsimp [Cj]; positivity |
| have hL : 0 ≤ L := by dsimp [L]; positivity |
| have hW : 0 < W := directedWeight_pos _ _ _ _ hedge |
| have hlH : lambda ≤ H := by |
| dsimp [H, weightH0] |
| exact le_add_of_nonneg_right (div_nonneg (by norm_num) (sub_pos.mpr hedge).le) |
| have hH : 0 < H := hlambda.trans_le hlH |
| have hCA : |C p - coordPartial 1 A p| ≤ (ell : ℝ) * MAs + MA * Mr := by |
| have he : C p - coordPartial 1 A p = |
| (ell : ℝ) * coordPartial 1 A p + A p * r p := by rw [hC]; ring |
| rw [he] |
| calc |
| _ ≤ |(ell : ℝ) * coordPartial 1 A p| + |A p * r p| := abs_add_le _ _ |
| _ = (ell : ℝ) * |coordPartial 1 A p| + |A p| * |r p| := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg ell)] |
| _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left hAs (Nat.cast_nonneg ell)) |
| (mul_le_mul hAp hr (abs_nonneg _) hMA) |
| have hT := primitive_directedT_lower_bound hB hR hp edge lambda epsilon MI MB |
| heps hedge hIs hBp hTbudget |
| have hJ0 := primitive_directedJ_bound hB hR hp A C hA edge lambda epsilon |
| ((ell : ℝ) * MAs + MA * Mr) MA MI heps hedge hH.le hCA hAp hIs |
| have hJ : |multiplierJ A B C (directedM edge lambda (coordinatePrimitive B)) |
| (directedN edge lambda epsilon (coordinatePrimitive B)) p| ≤ |
| W * Cj * epsilon * (1 + H * |A p|) := by |
| refine hJ0.trans ?_ |
| have hD : 0 ≤ (ell : ℝ) * MAs + MA * Mr + MA * MI := by positivity |
| have hh : 0 ≤ H * |A p| := mul_nonneg hH.le (abs_nonneg _) |
| have hbase : (ell : ℝ) * MAs + MA * Mr + MA * MI + H * |A p| ≤ |
| ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1) * (1 + H * |A p|) := by |
| nlinarith [mul_nonneg hD hh] |
| have hm := mul_le_mul_of_nonneg_left hbase (mul_nonneg (mul_nonneg hW.le heps) hR.le) |
| convert hm using 1 |
| dsimp only [W, H, Cj] |
| ring |
| have hcross := actual_normalized_cross_loss hW hH hCj heps hT hJ |
| have he : |directedSResidual B r ell epsilon p| ≤ E := |
| directedSResidual_bound heps heps1 hR.le (abs_lt.mpr hp.1).le hIs hBp hr |
| have hEH : E ≤ H / 8 := hEbudget.trans (by linarith) |
| have hLH : L / H ≤ mu / 4 := by |
| apply (div_le_iff₀ hH).2 |
| have hh : L ≤ (mu / 4) * lambda := (div_le_iff₀ hlambda).1 hloss |
| exact hh.trans (mul_le_mul_of_nonneg_left hlH (by positivity)) |
| have hmargin : mu / 2 ≤ directedPrincipal A ell epsilon p - H * A p / 2 + |
| A p * directedSResidual B r ell epsilon p - L * (H⁻¹ + H * (A p) ^ 2) := by |
| have hn : A p ≤ 0 := hAfar.trans (by linarith) |
| have hAa : alpha ≤ |A p| := by rw [abs_of_nonpos hn]; linarith |
| have hprincipal : -directedPrincipalBound ell epsilon R MAs MAt ≤ |
| directedPrincipal A ell epsilon p := |
| (abs_le.mp (directedPrincipal_abs_bound ell heps (abs_lt.mpr hp.1).le hAs hAt)).1 |
| have hbudget : directedPrincipalBound ell epsilon R MAs MAt + mu ≤ H * alpha / 4 := |
| hdominance.trans (div_le_div_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hlH halpha.le) (by norm_num)) |
| exact normalized_negative_far_coercivity hH hL hmu hn hAp hsmall hLH he hEH |
| hprincipal hAa hbudget |
| exact actual_quadratic_coercivity hW hH hlambda hmu hlH hT |
| (primitive_directedS_expansion hB hR hp A C r hA edge lambda epsilon ell hedge hC) |
| hcross hmargin |
|
|
| end SmoothLocal.Weighted |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
|
|
| theorem heightChartA_negative_of_curvature |
| (g : MetricField) (z : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (p : Coord) |
| {cG delta : ℝ} (hcG : 0 < cG) (hdelta : 0 < delta) |
| (hG : cG ≤ heightChartG1 g z Y p) |
| (hK : capPullback Y (gaussianCurvature g) p ≤ -delta) : |
| heightChartA g z Y p ≤ -(cG * delta) := by |
| rw [heightChartA_eq] |
| have hGpos : 0 ≤ heightChartG1 g z Y p := (hcG.trans_le hG).le |
| calc |
| _ ≤ heightChartG1 g z Y p * (-delta) := mul_le_mul_of_nonneg_left hK hGpos |
| _ ≤ cG * (-delta) := mul_le_mul_of_nonpos_right hG (by linarith) |
| _ = _ := by ring |
|
|
| end SmoothLocal.Flow |
|
|
| end |
|
|
| end OAI |
|
|
