| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.InnerCapStatistic |
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| namespace OAI |
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| open MeasureTheory Set Filter |
| noncomputable section |
| namespace CoulombAtom |
| lemma actual_inner_linear_statistic {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h a : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) |
| (hh : 0 < h) (ha : h ≤ a) {y : Space} (hy : 2*a ≤ ‖y‖) : |
| sourceField Z (actualInnerSource psi h) y*formMass psi ≤ |
| (h/a)*Real.sqrt (innerCapStatistic psi Z lam h)*Real.sqrt (formMass psi) := by |
| have hap : 0 < a := hh.trans_le ha |
| have hm := formMass_nonneg psi |
| by_cases hm0 : formMass psi = 0 |
| · simp only [hm0,mul_zero,Real.sqrt_zero,le_refl] |
| have hcap := actual_inner_exterior_cap hpsi hm0 hZ hlam hh (by linarith) |
| have hc := actualInnerBoundaryCap_nonneg psi Z hlam hh |
| have h1 : sourceField Z (actualInnerSource psi h) y ≤ actualInnerBoundaryCap psi Z lam h*(h/a) := by |
| calc |
| _ ≤ actualInnerBoundaryCap psi Z lam h*(2*h/‖y‖) := by simpa only [mul_div_assoc] using hcap |
| _ ≤ actualInnerBoundaryCap psi Z lam h*(2*h/(2*a)) := |
| mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_left (by positivity) (by positivity) hy) hc |
| _ = _ := by congr 1; field_simp |
| have hsq := actualInnerBoundaryCap_sq_le_statistic hpsi hZ hlam hh |
| have hstat := innerCapStatistic_nonneg psi Z lam h |
| have hroot : actualInnerBoundaryCap psi Z lam h*Real.sqrt (formMass psi) ≤ |
| Real.sqrt (innerCapStatistic psi Z lam h) := by |
| apply (sq_le_sq₀ (mul_nonneg hc (Real.sqrt_nonneg _)) (Real.sqrt_nonneg _)).mp |
| simpa only [mul_pow,Real.sq_sqrt hm,Real.sq_sqrt hstat] using hsq |
| have h2 := mul_le_mul_of_nonneg_right hroot (Real.sqrt_nonneg (formMass psi)) |
| rw [mul_assoc,←sq,Real.sq_sqrt hm] at h2 |
| calc |
| _ ≤ actualInnerBoundaryCap psi Z lam h*(h/a)*formMass psi := mul_le_mul_of_nonneg_right h1 hm |
| _ = (h/a)*(actualInnerBoundaryCap psi Z lam h*formMass psi) := by ring |
| _ ≤ (h/a)*(Real.sqrt (innerCapStatistic psi Z lam h)*Real.sqrt (formMass psi)) := |
| mul_le_mul_of_nonneg_left h2 (div_nonneg hh.le hap.le) |
| _ = _ := by ring |
| end CoulombAtom |
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| end |
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| end OAI |
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