openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /CoreObservationEnsembleBarrier.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Localization.CoreHistoryOutMaximumBarrier | |
| import OAI.Analysis.CoulombIonization.RadialBounds.CoupledCoreExcessBarrier | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators | |
| namespace CoulombAtom | |
| structure CoreObservationEnsemble where | |
| Branch : Type | |
| finite : Fintype Branch | |
| graph : Branch → CoreObservationGraph | |
| attribute [instance] CoreObservationEnsemble.finite | |
| namespace CoreObservationEnsemble | |
| def initial {N : ℕ} (ψ : FormVector N) (hψ : SobolevFermion ψ) : CoreObservationEnsemble where | |
| Branch := Unit | |
| finite := inferInstance | |
| graph := fun _ => CoreObservationGraph.initial ψ hψ | |
| def observe (E : CoreObservationEnsemble) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) : CoreObservationEnsemble where | |
| Branch := Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2 | |
| finite := inferInstance | |
| graph := fun ic => (E.graph ic.1).observe p hp ic.2 | |
| def mass (E : CoreObservationEnsemble) : ℝ := ∑ i, (E.graph i).mass | |
| def countMoment (E : CoreObservationEnsemble) (y : Space) (R : ℝ) : ℝ := | |
| ∑ i, (E.graph i).countMoment y R | |
| def fieldMoment (E : CoreObservationEnsemble) (Z lam : ℝ) (y : Space) : ℝ := | |
| ∑ i, (E.graph i).fieldMoment Z lam y | |
| def excess (E : CoreObservationEnsemble) (Z lam : ℝ) : ℝ := ∑ i, (E.graph i).excess Z lam | |
| def outMoment (E : CoreObservationEnsemble) (y : Space) {r b : ℝ} (hr : 0 ≤ r) (hb : 0 < b) | |
| (Z lam : ℝ) : ℝ := ∑ i, (E.graph i).outMoment y hr hb Z lam | |
| lemma mass_nonneg (E : CoreObservationEnsemble) : 0 ≤ E.mass := | |
| Finset.sum_nonneg (fun i _ => (E.graph i).mass_nonneg) | |
| lemma countMoment_nonneg (E : CoreObservationEnsemble) (y : Space) (R : ℝ) : 0 ≤ E.countMoment y R := | |
| Finset.sum_nonneg (fun i _ => (E.graph i).countMoment_nonneg y R) | |
| lemma fieldMoment_nonneg (E : CoreObservationEnsemble) (Z lam : ℝ) (y : Space) : 0 ≤ E.fieldMoment Z lam y := | |
| Finset.sum_nonneg (fun i _ => (E.graph i).fieldMoment_nonneg Z lam y) | |
| lemma outMoment_nonneg (E : CoreObservationEnsemble) (y : Space) {r b : ℝ} (hr : 0 ≤ r) (hb : 0 < b) | |
| (Z lam : ℝ) : 0 ≤ E.outMoment y hr hb Z lam := | |
| Finset.sum_nonneg (fun i _ => (E.graph i).outMoment_nonneg y hr hb Z lam) | |
| lemma observe_mass (E : CoreObservationEnsemble) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) : (E.observe p hp).mass = E.mass := by | |
| change (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, ((E.graph ic.1).observe p hp ic.2).mass) = _ | |
| rw [Fintype.sum_sigma] | |
| simp_rw [CoreObservationGraph.observe_mass] | |
| rfl | |
| lemma observe_countMoment (E : CoreObservationEnsemble) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) (y : Space) (R : ℝ) : | |
| (E.observe p hp).countMoment y R ≤ E.countMoment y R := by | |
| change (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, | |
| ((E.graph ic.1).observe p hp ic.2).countMoment y R) ≤ _ | |
| rw [Fintype.sum_sigma] | |
| exact Finset.sum_le_sum (fun i _ => (E.graph i).observe_countMoment p hp y R) | |
| lemma fieldMoment_le_nuclear (E : CoreObservationEnsemble) {Z lam : ℝ} | |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (y : Space) : E.fieldMoment Z lam y ≤ (Z/‖y‖)^2*E.mass := by | |
| unfold fieldMoment mass | |
| rw [Finset.mul_sum] | |
| exact Finset.sum_le_sum (fun i _ => (E.graph i).fieldMoment_le_nuclear hZ hlam y) | |
| lemma fieldMoment_ball_integrable (E : CoreObservationEnsemble) (y : Space) {A d Z lam : ℝ} | |
| (hd : 0 < d) (hs : A+d ≤ ‖y‖) (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) : | |
| IntegrableOn (E.fieldMoment Z lam) (closedBall y A) := | |
| integrable_finsetSum _ (fun i _ => coreLawAverage_fieldSquare_ball_integrable | |
| (E.graph i).sobolev y hd hs hZ hlam) | |
| lemma observe_excess_budget (E : CoreObservationEnsemble) (y : Space) {r b R : ℝ} | |
| (hr : 0 ≤ r) (hb : 0 < b) (hy : r+2*b ≤ ‖y‖) (hR : r+b < R) | |
| {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) : | |
| (E.observe (coreFirstRadialCut y hr hb) (coreFirstRadialCut_partition y hr hb)).excess Z lam ≤ | |
| E.excess Z lam+((3/2:ℝ)*(Real.pi*smoothTransitionBound/b)^2)* | |
| (Real.sqrt (E.countMoment y R)*Real.sqrt E.mass)+ | |
| Real.sqrt (E.outMoment y hr hb Z lam)*Real.sqrt (E.countMoment y R) := by | |
| let K := (3/2:ℝ)*(Real.pi*smoothTransitionBound/b)^2 | |
| have h1 := Real.sum_sqrt_mul_sqrt_le Finset.univ | |
| (fun i : E.Branch => (E.graph i).countMoment_nonneg y R) (fun i => (E.graph i).mass_nonneg) | |
| have h2 := Real.sum_sqrt_mul_sqrt_le Finset.univ | |
| (fun i : E.Branch => (E.graph i).outMoment_nonneg y hr hb Z lam) | |
| (fun i => (E.graph i).countMoment_nonneg y R) | |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i (_ : i ∈ Finset.univ) => | |
| (E.graph i).observe_excess_budget y hr hb hy hR hZ hlam) | |
| simp only [Finset.sum_add_distrib,←Finset.mul_sum] at hh | |
| have hend := hh.trans (add_le_add (add_le_add le_rfl (mul_le_mul_of_nonneg_left h1 (by positivity : 0 ≤ K))) h2) | |
| change (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, | |
| ((E.graph ic.1).observe (coreFirstRadialCut y hr hb) (coreFirstRadialCut_partition y hr hb) ic.2).excess Z lam) ≤ _ | |
| rw [Fintype.sum_sigma] | |
| exact hend | |
| lemma fieldMoment_le_observe_excess (E : CoreObservationEnsemble) | |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (hp : ∀ x, ∑ a, (p a).value x^2 = 1) | |
| (y : Space) {B : ℝ} (hB : 0 < B) (hnuc : 2*B ≤ ‖y‖) | |
| (hv : ∀ x ∈ ball y (2*B), (p 0).value x = 0) | |
| (hd : ∀ x ∈ ball y (2*B), ∀ a, lineDeriv ℝ (p 0).value x (spaceDirections a) = 0) | |
| {Z lam : ℝ} (hZ : 0 ≤ Z) (hlam : 0 < lam) : | |
| E.fieldMoment Z lam y ≤ 2*packetFieldConstant^2*((1/B^4+1/B)^2*E.mass+ | |
| (E.observe p hp).excess Z lam/B) := by | |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i (_ : i ∈ Finset.univ) => | |
| (E.graph i).fieldMoment_le_observe_excess p hp y hB hnuc hv hd hZ hlam) | |
| simp only [←Finset.mul_sum,Finset.sum_add_distrib,←Finset.sum_div] at hh | |
| change E.fieldMoment Z lam y ≤ 2*packetFieldConstant^2*((1/B^4+1/B)^2*E.mass+ | |
| (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, | |
| ((E.graph ic.1).observe p hp ic.2).excess Z lam)/B) | |
| rw [Fintype.sum_sigma] | |
| exact hh | |
| lemma outMoment_spatial (E : CoreObservationEnsemble) (y : Space) {r b R B Z lam : ℝ} | |
| (hr : 0 ≤ r) (hb : 0 < b) (hR : 0 < R) (hsep : r+b+2*R ≤ ‖y‖) | |
| (hnear : r+2*b ≤ ‖y‖) (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) {η : Space → ℝ} | |
| (hm : Measurable η) (hn : ∀ z, 0 ≤ η z) (hs : ∀ z, η z ≠ 0 → ‖z‖ ≤ R) | |
| (hbound : ∀ z, |η z| ≤ B) (hηr : CoulombAnalysis.IsRadial η) (h1 : ∫ z, η z = 1) : | |
| E.outMoment y hr hb Z lam ≤ B*∫ z in closedBall y (r+b+R), | |
| (E.observe (coreFirstRadialCut y hr hb) (coreFirstRadialCut_partition y hr hb)).fieldMoment Z lam z := by | |
| let p := coreFirstRadialCut y hr hb | |
| let hp := coreFirstRadialCut_partition y hr hb | |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun i (_ : i ∈ Finset.univ) => | |
| (E.graph i).outMoment_spatial y hr hb hR hsep hnear hZ hlam hm hn hs hbound hηr h1) | |
| rw [←Finset.mul_sum] at hh | |
| change _ ≤ B*∫ z in closedBall y (r+b+R), | |
| ∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, | |
| ((E.graph ic.1).observe p hp ic.2).fieldMoment Z lam z | |
| simp_rw [Fintype.sum_sigma] | |
| rw [integral_finsetSum] | |
| · exact hh | |
| · intro i _ | |
| exact integrable_finsetSum _ (fun c _ => coreLawAverage_fieldSquare_ball_integrable | |
| ((E.graph i).observe p hp c).sobolev y hR (by linarith) hZ hlam) | |
| end CoreObservationEnsemble | |
| end CoulombAtom | |
| end | |
| end OAI | |