openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /EventLawSelectedMaster.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Variational.SelectedMasterComparison | |
| import OAI.Analysis.CoulombIonization.Localization.OrdinaryDensity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators ENNReal ContDiff | |
| namespace CoulombAtom | |
| open CoulombAnalysis CoulombObservation | |
| theorem eventLaw_selected_master_density {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) | |
| {N K : ℕ} (F G : fermionGraph N) (hG : ‖fermionGraphValue N G‖^2 = 1) | |
| (ell : Fin K → ℝ) (j : ℕ) {A : Set (Fin K × (Fin N × Fin 3) → ℝ)} | |
| (hinfo : MeasurableSet[observationInformation ell j] (physicalObservationEvent ell A)) | |
| (hp : 0 < physicalObservationProbability F ell A) | |
| (hlaw : graphRawLaw G = (ENNReal.ofReal (physicalObservationProbability F ell A))⁻¹ • | |
| Measure.map Prod.fst ((physicalObservationLaw (graphRawLaw F) K).restrict | |
| (physicalObservationEvent ell A))) | |
| (y : Space) {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) | |
| (hr : 0 < r₀) (hs : 0 < s) (hs1 : s ≤ 1) | |
| {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g) | |
| (hgn : ∫ z, (g z)^2 = 1) (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) | |
| {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hba : 7*b < 5*a) (hy : 6*a+2*b ≤ ‖y‖) | |
| {q : ℝ} (hq : 0 < q) (hqR : q ≤ 3*(5*a-4*b)/4) : | |
| ∃ t ∈ Icc (5*a) (6*a), ∃ ht : 0 ≤ t, | |
| |Z/‖y‖-lam-(physicalObservationProbability F ell A)⁻¹* | |
| (∫ z in physicalObservationEvent ell A, | |
| tfPotential (jointMasterPosterior (graphRawLaw F) ell j c₁ r₀ s g (originalDatum ell j z)) y | |
| ∂physicalObservationLaw (graphRawLaw F) K)- | |
| expectedRadialPatchCenter (graphFormVector G) y ht hb Z lam| ≤ | |
| selectedPotentialError (graphFormVector G) y a ht hb Z lam q g+ | |
| (∫ z in ball y (2*masterWidth c₁ r₀ s y), | |
| ordinaryDensity (graphFormVector G) z/‖z-y‖) := by | |
| have hmass : formMass (graphFormVector G) = 1 := (formMass_graph G).trans hG | |
| obtain ⟨t,ht,ht0,herr⟩ := exists_selected_radial_potential_error | |
| (graphFormVector_sobolev G) hmass y ha hb hba hy hZ hlam hg hcg hgn hrad hgs hq | |
| refine ⟨t,ht,ht0,?_⟩ | |
| have hi : Integrable (rawPotential y) (graphRawLaw F) := by | |
| rw [←formRawLaw_graph] | |
| exact rawPotential_form_integrable (graphFormVector_sobolev F).sobolevVector y | |
| have he : ∀ᵐ x ∂graphRawLaw F, ∀ i, x i ≠ y := by | |
| rw [←formRawLaw_graph] | |
| exact formRawLaw_ae_no_poles (graphFormVector F) y | |
| have hh := original_master_radial_center_comparison (graphRawLaw F) (graphFormVector_sobolev G) | |
| ell j y hi he hc hcL hr hs hs1 hg hcg hgn hrad hgs hinfo hp | |
| (by rw [formRawLaw_graph]; exact hlaw) ht0 hb (by linarith [ht.1]) | |
| (by linarith [ht.2]) hZ hlam hq (by linarith [ht.1]) | |
| have hh' := hh.trans (add_le_add herr le_rfl) | |
| have hd := ordinaryDensity_raw_local_potential | |
| (graphFormVector_sobolev G).sobolevVector y (2*masterWidth c₁ r₀ s y) | |
| simpa only [hmass,mul_one,hd] using hh' | |
| end CoulombAtom | |
| end | |
| end OAI | |