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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 | import OAI.Geometry.IsometricImmersion.Metrics.ModelProfile
import OAI.Geometry.IsometricImmersion.Energy.DirectedCoercivity
import Mathlib.Analysis.Calculus.LocalExtr.Basic
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Model
open SmoothLocal.Geometry SmoothLocal.Weighted
theorem squareTransition_deriv_zero_left {t : ℝ} (ht : t ≤ 1 / 4) :
deriv squareTransition t = 0 := by
have hmin : IsLocalMin squareTransition t := Filter.Eventually.of_forall (fun x => by
rw [squareTransition_zero ht]
exact (squareTransition_range x).1)
exact hmin.deriv_eq_zero
theorem squareTransition_deriv_zero_right {t : ℝ} (ht : 1 / 2 ≤ t) :
deriv squareTransition t = 0 := by
have hmax : IsLocalMax squareTransition t := Filter.Eventually.of_forall (fun x => by
rw [squareTransition_one ht]
exact (squareTransition_range x).2)
exact hmax.deriv_eq_zero
theorem modelProfile_deriv_outer {y : ℝ} (hy : 1 / 2 ≤ y ^ 2) :
deriv modelProfile y = -2 * y := by
rw [modelProfile_deriv, squareTransition_one hy, squareTransition_deriv_zero_right hy]
ring
theorem modelProfile_deriv_nonneg_lower_cap {y : ℝ} (hy : y ≤ 1 / 50) :
0 ≤ deriv modelProfile y := by
by_cases hy0 : y ≤ 0
· rw [modelProfile_deriv]
have hy3 : y ^ 3 ≤ 0 := by
have hh := mul_nonpos_of_nonpos_of_nonneg hy0 (sq_nonneg y)
nlinarith only [hh]
have hfirst : 0 ≤ -2 * y * squareTransition (y ^ 2) :=
mul_nonneg (by linarith) (squareTransition_range _).1
have hsecond : 0 ≤ -2 * y ^ 3 * deriv squareTransition (y ^ 2) :=
mul_nonneg (by linarith) (squareTransition_deriv_nonneg _)
linarith
· have hypos : 0 < y := lt_of_not_ge hy0
have hysq : y ^ 2 ≤ 1 / 4 :=
(pow_le_pow_left₀ hypos.le hy 2).trans (by norm_num)
rw [modelProfile_deriv, squareTransition_zero hysq, squareTransition_deriv_zero_left hysq]
norm_num
theorem modelProfile_deriv_lower_three_quarters {y : ℝ} (hy : y ≤ -(3 / 4 : ℝ)) :
3 / 2 ≤ deriv modelProfile y := by
have hs : (3 / 4 : ℝ) ^ 2 ≤ (-y) ^ 2 :=
pow_le_pow_left₀ (by norm_num) (by linarith : (3 / 4 : ℝ) ≤ -y) 2
have hysq : 1 / 2 ≤ y ^ 2 := by nlinarith only [hs]
rw [modelProfile_deriv_outer hysq]
linarith
theorem modelProfile_middle_lower {y : ℝ}
(hylo : -(3 / 4 : ℝ) ≤ y) (hyhi : y ≤ 1 / 50) :
7 / 16 ≤ modelProfile y := by
have hy2 : y ^ 2 ≤ (3 / 4 : ℝ) ^ 2 := sq_le_sq' hylo (by linarith)
nlinarith [(modelProfile_bounds y).1]
theorem modelCurvature_partial (kappa : ℝ) (p : Coord) (i : Fin 2) :
coordPartial i (modelCurvature kappa) p =
if i = 0 then 2 * kappa * p 0 else -kappa * deriv modelProfile (p 1) := by
have hx := hasFDerivAt_apply (𝕜 := ℝ) 0 p
have hy := ((modelProfile_contDiff.differentiable (by simp) (p 1)).hasDerivAt).comp_hasFDerivAt
p (hasFDerivAt_apply (𝕜 := ℝ) 1 p)
have hd := ((hx.pow 2).sub hy).const_mul kappa
simp only [Function.comp_def, Pi.sub_def] at hd
change fderiv ℝ (fun q : Coord => kappa * ((q 0) ^ 2 - modelProfile (q 1))) p
(Pi.single i 1) = _
rw [hd.fderiv]
fin_cases i <;> simp
ring
theorem modelCurvature_partial_y_nonpos {kappa : ℝ} (hkappa : 0 ≤ kappa)
{p : Coord} (hy : p 1 ≤ 1 / 50) : coordPartial 1 (modelCurvature kappa) p ≤ 0 := by
rw [modelCurvature_partial]
norm_num only [Fin.isValue, Fin.reduceEq, ↓reduceIte]
exact mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hkappa)
(modelProfile_deriv_nonneg_lower_cap hy)
theorem modelCurvature_negative_y_slope {kappa : ℝ} (hkappa : 0 ≤ kappa)
{p : Coord} (hy : p 1 ≤ -(3 / 4 : ℝ)) :
3 * kappa / 2 ≤ -coordPartial 1 (modelCurvature kappa) p := by
rw [modelCurvature_partial]
rw [ite_eq_right (by decide : (1 : Fin 2) ≠ 0)]
have hm := mul_le_mul_of_nonneg_left (modelProfile_deriv_lower_three_quarters hy) hkappa
nlinarith only [hm]
theorem modelCurvature_middle_xsq_lower {kappa : ℝ} (hkappa : 0 < kappa)
{p : Coord} (hylo : -(3 / 4 : ℝ) ≤ p 1) (hyhi : p 1 ≤ 1 / 50)
(hK : |modelCurvature kappa p| ≤ kappa / 8) :
5 / 16 ≤ (p 0) ^ 2 := by
have hh := mul_le_mul_of_nonneg_left (modelProfile_middle_lower hylo hyhi) hkappa.le
have hlower := (abs_le.mp hK).1
unfold modelCurvature at hlower
have hprod : kappa * (5 / 16) ≤ kappa * (p 0) ^ 2 := by nlinarith only [hh, hlower]
exact le_of_mul_le_mul_left hprod hkappa
def modelCoercivityMargin (g0 kappa rhoMax epsilon : ℝ) (ell : ℕ) : ℝ :=
g0 * kappa * min (3 * ((ell : ℝ) + 1 / 2) / (4 * rhoMax)) (5 * epsilon / 16)
theorem modelCoercivityMargin_pos {g0 kappa rhoMax epsilon : ℝ} (ell : ℕ)
(hg0 : 0 < g0) (hkappa : 0 < kappa) (hrho : 0 < rhoMax) (heps : 0 < epsilon) :
0 < modelCoercivityMargin g0 kappa rhoMax epsilon ell := by
have hell : 0 ≤ (ell : ℝ) := Nat.cast_nonneg ell
unfold modelCoercivityMargin
positivity
theorem directed_model_slope_lower {rho rhoMax epsilon R Mq t q : ℝ} (ell : ℕ)
(hrho : 0 < rho) (hrhoMax : rho ≤ rhoMax) (heps : 0 ≤ epsilon)
(ht : |t| ≤ R) (hq : |q| ≤ Mq)
(hsmall : epsilon * R * Mq ≤ ((ell : ℝ) + 1 / 2) / rhoMax) :
((ell : ℝ) + 1 / 2) / (2 * rhoMax) ≤
((ell : ℝ) + 1 / 2) / rho + epsilon * t * q / 2 := by
have hR : 0 ≤ R := (abs_nonneg _).trans ht
have hMq : 0 ≤ Mq := (abs_nonneg _).trans hq
have hmax : 0 < rhoMax := hrho.trans_le hrhoMax
have hbeta : 0 ≤ (ell : ℝ) + 1 / 2 := by positivity
have hfrac : ((ell : ℝ) + 1 / 2) / rhoMax ≤ ((ell : ℝ) + 1 / 2) / rho :=
(div_le_div_iff₀ hmax hrho).2 (mul_le_mul_of_nonneg_left hrhoMax hbeta)
have hprod : |epsilon * t * q| ≤ epsilon * R * Mq := by
rw [abs_mul, abs_mul, abs_of_nonneg heps]
exact mul_le_mul (mul_le_mul_of_nonneg_left ht heps) hq
(abs_nonneg _) (mul_nonneg heps hR)
have hprodlo := (abs_le.mp (hprod.trans hsmall)).1
have he : ((ell : ℝ) + 1 / 2) / (2 * rhoMax) =
(((ell : ℝ) + 1 / 2) / rhoMax) / 2 := by ring
rw [he]
linarith
theorem model_geometric_principal_lower
{kappa g0 G rho rhoMax epsilon R Mq q : ℝ} {p : Coord} (ell : ℕ)
(hkappa : 0 < kappa) (hg0 : 0 < g0) (hG : g0 ≤ G)
(hrho : 0 < rho) (hrhoMax : rho ≤ rhoMax) (heps : 0 < epsilon)
(ht : |p 0| ≤ R) (hq : |q| ≤ Mq)
(hsmall : epsilon * R * Mq ≤ ((ell : ℝ) + 1 / 2) / rhoMax)
(hy : p 1 ≤ 1 / 50) (hK : |modelCurvature kappa p| ≤ kappa / 8) :
modelCoercivityMargin g0 kappa rhoMax epsilon ell ≤
G * (-(((ell : ℝ) + 1 / 2) / rho + epsilon * p 0 * q / 2) *
coordPartial 1 (modelCurvature kappa) p + epsilon * kappa * (p 0) ^ 2) := by
let beta : ℝ := (ell : ℝ) + 1 / 2
let coef : ℝ := beta / rho + epsilon * p 0 * q / 2
let amin : ℝ := beta / (2 * rhoMax)
let base : ℝ := coef * (-coordPartial 1 (modelCurvature kappa) p) + epsilon * kappa * (p 0) ^ 2
have hmax : 0 < rhoMax := hrho.trans_le hrhoMax
have hbeta : 0 < beta := by dsimp [beta]; positivity
have hamin : 0 < amin := by dsimp [amin]; positivity
have hcoef : amin ≤ coef := directed_model_slope_lower ell hrho hrhoMax heps.le ht hq hsmall
have hKy : 0 ≤ -coordPartial 1 (modelCurvature kappa) p :=
neg_nonneg.mpr (modelCurvature_partial_y_nonpos hkappa.le hy)
have hterm : 0 ≤ epsilon * kappa * (p 0) ^ 2 := by positivity
have hbase : 0 ≤ base := by dsimp [base]; exact add_nonneg (mul_nonneg (hamin.le.trans hcoef) hKy) hterm
have hlower : kappa * min (3 * beta / (4 * rhoMax)) (5 * epsilon / 16) ≤ base := by
by_cases hylo : p 1 ≤ -(3 / 4 : ℝ)
· have hslope := modelCurvature_negative_y_slope hkappa.le hylo
calc
_ ≤ kappa * (3 * beta / (4 * rhoMax)) := mul_le_mul_of_nonneg_left (min_le_left _ _) hkappa.le
_ = amin * (3 * kappa / 2) := by dsimp [amin]; ring
_ ≤ amin * (-coordPartial 1 (modelCurvature kappa) p) :=
mul_le_mul_of_nonneg_left hslope hamin.le
_ ≤ coef * (-coordPartial 1 (modelCurvature kappa) p) := mul_le_mul_of_nonneg_right hcoef hKy
_ ≤ base := by dsimp [base]; linarith
· have hxsq := modelCurvature_middle_xsq_lower hkappa (lt_of_not_ge hylo).le hy hK
calc
_ ≤ kappa * (5 * epsilon / 16) := mul_le_mul_of_nonneg_left (min_le_right _ _) hkappa.le
_ = epsilon * kappa * (5 / 16) := by ring
_ ≤ epsilon * kappa * (p 0) ^ 2 := mul_le_mul_of_nonneg_left hxsq (mul_pos heps hkappa).le
_ ≤ base := by
dsimp [base]
have hn := mul_nonneg (hamin.le.trans hcoef) hKy
linarith
have hfirst := mul_le_mul_of_nonneg_left hlower hg0.le
have hsecond := mul_le_mul_of_nonneg_right hG hbase
unfold modelCoercivityMargin
dsimp only [base, coef, beta] at hfirst hsecond
nlinarith only [hfirst, hsecond]
end SmoothLocal.Model
end
end OAI
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