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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 | import OAI.Geometry.IsometricImmersion.Darboux.ModelPrincipal
import OAI.Geometry.IsometricImmersion.Energy.DirectedFarNegative
namespace OAI
noncomputable section
open scoped ContDiff
namespace SmoothLocal.Weighted
open SmoothLocal.Model
theorem exists_directed_parameters (ell : ℕ)
{kappa cG rhoMax MG MI MB Mr MA MAs MAt MG1 D : ℝ}
(hk : 0 < kappa) (hcG : 0 < cG) (hrho : 0 < rhoMax)
(hMG : 0 ≤ MG) (hMI : 0 ≤ MI) (hMB : 0 ≤ MB) (hMr : 0 ≤ Mr)
(hMA : 0 ≤ MA) (hMAs : 0 ≤ MAs) (hMAt : 0 ≤ MAt) (hMG1 : 0 ≤ MG1) (hD : 0 ≤ D) :
∃ epsilon lambda c1 : ℝ,
0 < epsilon ∧ epsilon ≤ 1 ∧ 0 < lambda ∧ 0 < c1 ∧
epsilon * 2 * ((1 : ℝ) / 100) ≤ ((ell : ℝ) + 1 / 2) / rhoMax ∧
2 * (MI + epsilon + epsilon * 2 * MB) ≤ lambda ∧
(MI + 1 + 2 * MB) / 2 + Mr ≤ lambda / 8 ∧
4 * (2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1))^2 * epsilon^2 * MA ≤ 1 / 8 ∧
(4 * (2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1))^2 * epsilon^2) / lambda ≤
modelCoercivityMargin cG kappa rhoMax epsilon ell / 8 ∧
directedPrincipalBound ell epsilon 2 MAs MAt + modelCoercivityMargin cG kappa rhoMax epsilon ell / 2 ≤
lambda * (cG * modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell) / 4 ∧
c1 * D ≤ modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell ∧
MG1 * c1 * (lambda * D + 8) ≤ modelCoercivityMargin cG kappa rhoMax epsilon ell / 6 := by
let Cj := 2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1)
let L := 4 * Cj^2
have hCj : 0 ≤ Cj := by dsimp [Cj]; positivity
have hL : 0 ≤ L := by dsimp [L]; positivity
have hbeta : 0 < ((ell : ℝ) + 1 / 2) / rhoMax := by positivity
have hden : 0 < 8 * L * MA + 1 := by positivity
let epsilon := min 1 (min (((ell : ℝ) + 1 / 2) / rhoMax) (1 / (8 * L * MA + 1)))
have heps : 0 < epsilon := by dsimp [epsilon]; positivity
have heps1 : epsilon ≤ 1 := min_le_left _ _
have hepsbeta : epsilon ≤ ((ell : ℝ) + 1 / 2) / rhoMax :=
(min_le_right _ _).trans (min_le_left _ _)
have hepsden : epsilon ≤ 1 / (8 * L * MA + 1) :=
(min_le_right _ _).trans (min_le_right _ _)
have hcross : L * epsilon^2 * MA ≤ 1 / 8 := by
have hm := (le_div_iff₀ hden).mp hepsden
have hsquare : epsilon^2 ≤ epsilon := by nlinarith only [heps.le, heps1]
have hprod := mul_le_mul_of_nonneg_left hsquare (mul_nonneg hL hMA)
nlinarith only [hm, hprod, heps.le]
have hqbudget : epsilon * 2 * ((1 : ℝ) / 100) ≤ ((ell : ℝ) + 1 / 2) / rhoMax := by
have hh : epsilon * 2 * ((1 : ℝ) / 100) ≤ epsilon := by linarith only [heps.le]
exact hh.trans hepsbeta
let margin := modelCoercivityMargin cG kappa rhoMax epsilon ell
let delta := modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell
let alpha := cG * delta
have hmargin : 0 < margin := modelCoercivityMargin_pos ell hcG hk hrho heps
have hdelta : 0 < delta := modelPrincipalRadius_pos ell hcG hk hrho heps (by norm_num) hMG
have halpha : 0 < alpha := mul_pos hcG hdelta
let T0 := 2 * (MI + 1 + 2 * MB)
let E0 := (MI + 1 + 2 * MB) / 2 + Mr
let P0 := ((ell : ℝ) + 1 / 2) * MAs + MAt
have hT0 : 0 ≤ T0 := by dsimp [T0]; positivity
have hE0 : 0 ≤ E0 := by dsimp [E0]; positivity
have hP0 : 0 ≤ P0 := by dsimp [P0]; positivity
let lambda := 1 + T0 + 8 * E0 + 8 * L * epsilon^2 / margin + 4 * (P0 + margin / 2) / alpha
have hLterm : 0 ≤ 8 * L * epsilon^2 / margin := by positivity
have hFterm : 0 ≤ 4 * (P0 + margin / 2) / alpha := by positivity
have hlambda : 0 < lambda := by dsimp [lambda]; linarith only [hT0, hE0, hLterm, hFterm]
have hTlambda : T0 ≤ lambda := by dsimp [lambda]; linarith only [hE0, hLterm, hFterm]
have hElambda : 8 * E0 ≤ lambda := by dsimp [lambda]; linarith only [hT0, hLterm, hFterm]
have hLlambda : 8 * L * epsilon^2 / margin ≤ lambda := by dsimp [lambda]; linarith only [hT0, hE0, hFterm]
have hFlambda : 4 * (P0 + margin / 2) / alpha ≤ lambda := by dsimp [lambda]; linarith only [hT0, hE0, hLterm]
have hTbudget : 2 * (MI + epsilon + epsilon * 2 * MB) ≤ lambda := by
have hp := mul_le_mul_of_nonneg_right heps1 (show 0 ≤ 2 * MB by positivity)
have hh : 2 * (MI + epsilon + epsilon * 2 * MB) ≤ T0 := by dsimp [T0]; nlinarith only [hp, heps1]
exact hh.trans hTlambda
have hEbudget : E0 ≤ lambda / 8 := by linarith only [hElambda]
have hloss : L * epsilon^2 / lambda ≤ margin / 8 := by
have hh := (div_le_iff₀ hmargin).mp hLlambda
apply (div_le_iff₀ hlambda).mpr
nlinarith only [hh]
have hprincipal : directedPrincipalBound ell epsilon 2 MAs MAt ≤ P0 := by
have hh := mul_le_mul_of_nonneg_right heps1 hMAt
dsimp [directedPrincipalBound, P0]
nlinarith only [hh]
have hfar : directedPrincipalBound ell epsilon 2 MAs MAt + margin / 2 ≤ lambda * alpha / 4 := by
have hh := (div_le_iff₀ halpha).mp hFlambda
nlinarith only [hh, hprincipal]
have hD1 : 0 < D + 1 := by linarith only [hD]
have hstripden : 0 < 6 * (MG1 + 1) * (lambda * D + 8) := by positivity
let c1 := min (delta / (D + 1)) (margin / (6 * (MG1 + 1) * (lambda * D + 8)))
have hc1 : 0 < c1 := by dsimp [c1]; positivity
have hc1delta : c1 ≤ delta / (D + 1) := min_le_left _ _
have hc1strip : c1 ≤ margin / (6 * (MG1 + 1) * (lambda * D + 8)) := min_le_right _ _
have hnearstrip : c1 * D ≤ delta := by
have hh := (le_div_iff₀ hD1).mp hc1delta
nlinarith only [hh, hc1.le]
have hstrip : MG1 * c1 * (lambda * D + 8) ≤ margin / 6 := by
have hh := (le_div_iff₀ hstripden).mp hc1strip
have hfac : 0 ≤ c1 * (lambda * D + 8) := by positivity
have hbase : MG1 * c1 * (lambda * D + 8) ≤ (MG1 + 1) * c1 * (lambda * D + 8) := by
nlinarith only [hfac]
have hfull : (MG1 + 1) * c1 * (lambda * D + 8) ≤ margin / 6 := by nlinarith only [hh]
exact hbase.trans hfull
exact ⟨epsilon, lambda, c1, heps, heps1, hlambda, hc1, hqbudget, hTbudget,
hEbudget, hcross, hloss, hfar, hnearstrip, hstrip⟩
end SmoothLocal.Weighted
end
end OAI
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