| import OAI.Geometry.IsometricImmersion.Darboux.ModelPrincipal |
| import OAI.Geometry.IsometricImmersion.Energy.DirectedFarNegative |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff |
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| namespace SmoothLocal.Weighted |
| open SmoothLocal.Model |
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| 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⟩ |
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| end SmoothLocal.Weighted |
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| end |
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| end OAI |
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