| import Mathlib |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.InnerFieldCap |
| import OAI.Analysis.CoulombIonization.RadialBounds.CoulombTailStatistic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace CoulombAtom |
|
|
| section |
| open MeasureTheory Filter Set |
| open scoped BigOperators |
|
|
| def innerFieldVolume (h : ℝ) : ℝ := volume.real (innerFieldTestAnnulus h) |
| lemma innerFieldVolume_nonneg (h : ℝ) : 0 ≤ innerFieldVolume h := measureReal_nonneg |
|
|
| lemma positiveCoreField_annular_sq_integrable {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| IntegrableOn (fun z => (max (normalizedCoreField Z lam psi z) 0)^2) (innerFieldTestAnnulus h) := by |
| let : IsFiniteMeasure (volume.restrict (innerFieldTestAnnulus h)) := |
| ⟨by rw [Measure.restrict_apply_univ]; exact innerFieldTestAnnulus_finite h⟩ |
| apply (integrable_const ((Z/h)^2)).mono' |
| (((normalizedCoreField_measurable hpsi Z lam).max measurable_const).pow_const 2).aestronglyMeasurable |
| filter_upwards [ae_restrict_mem (innerFieldTestAnnulus_measurable h)] with z hz |
| rw [Real.norm_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (le_max_right _ _) (normalizedCoreField_pos_le psi hZ hlam hh hz.1) 2 |
|
|
| lemma positiveCoreField_annular_cauchy {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| (∫ z in innerFieldTestAnnulus h, max (normalizedCoreField Z lam psi z) 0)^2 ≤ |
| innerFieldVolume h*(∫ z in innerFieldTestAnnulus h, (max (normalizedCoreField Z lam psi z) 0)^2) := by |
| let mu := volume.restrict (innerFieldTestAnnulus h) |
| let : IsFiniteMeasure mu := ⟨by rw [Measure.restrict_apply_univ]; exact innerFieldTestAnnulus_finite h⟩ |
| let f := fun z => max (normalizedCoreField Z lam psi z) 0 |
| have h1 : Integrable f mu := positiveCoreField_annular_integrable hpsi hZ hlam hh |
| have h2 : Integrable (fun z => f z^2) mu := positiveCoreField_annular_sq_integrable hpsi hZ hlam hh |
| have hc := weighted_sum_integral_cauchy (fun _ : Unit => mu) |
| (fun _ _ => (1:ℝ)) (fun _ z => f z) (fun _ _ => (1:ℝ)) |
| (fun _ _ => zero_le_one) |
| (fun _ => by simpa only [mul_one] using h2) |
| (fun _ => by simpa only [one_pow,one_mul] using (integrable_const (1:ℝ) : Integrable (fun _ : Space => (1:ℝ)) mu)) |
| (fun _ => by simpa only [mul_one] using h1) |
| simp only [Finset.univ_unique,Finset.sum_singleton,mul_one,one_pow,integral_const, |
| smul_eq_mul,Measure.real] at hc |
| have hsq := pow_le_pow_left₀ (integral_nonneg (fun z => le_max_right _ _)) hc 2 |
| rw [mul_pow,Real.sq_sqrt (integral_nonneg (fun z => sq_nonneg _)), |
| Real.sq_sqrt (by dsimp [mu]; positivity)] at hsq |
| simpa only [f,mu,innerFieldVolume,Measure.real,Measure.restrict_apply_univ,mul_comm] using hsq |
|
|
| def innerCapStatistic {N : ℕ} (psi : FormVector N) (Z lam h : ℝ) : ℝ := |
| 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h* |
| (∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z psi)+ |
| 3*lam^2*formMass psi+3*(16*packetPotentialConstant)^2*rawWeightedMoment psi (coulombTailWeight h) |
|
|
| lemma innerCapStatistic_nonneg {N : ℕ} (psi : FormVector N) (Z lam h : ℝ) : |
| 0 ≤ innerCapStatistic psi Z lam h := by |
| unfold innerCapStatistic |
| have hv := innerFieldVolume_nonneg h |
| have hf : 0 ≤ ∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z psi := |
| integral_nonneg (fun z => coreFieldSquare_nonneg Z lam z psi) |
| have hm := formMass_nonneg psi |
| have ht := rawWeightedMoment_nonneg psi (coulombTailWeight h) |
| positivity |
|
|
| theorem actualInnerBoundaryCap_sq_le_statistic {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) {Z lam h : ℝ} (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| (actualInnerBoundaryCap psi Z lam h)^2*formMass psi ≤ innerCapStatistic psi Z lam h := by |
| let A := packetDensityConstant/(h/4)^3 |
| let B := 16*packetPotentialConstant |
| let U := ∫ z in innerFieldTestAnnulus h, max (normalizedCoreField Z lam psi z) 0 |
| let T := ∫ a, ‖a‖⁻¹ ∂(actualParticleSource psi).restrict {a | h < ‖a‖} |
| let M := formMass psi |
| have hM : 0 ≤ M := formMass_nonneg psi |
| have hU := mul_le_mul_of_nonneg_right (positiveCoreField_annular_cauchy hpsi hZ hlam hh) hM |
| have hT := actual_tail_square_le hpsi hh |
| have hx : (A*U+lam+B*T)^2 ≤ 3*((A*U)^2+lam^2+(B*T)^2) := by |
| nlinarith only [sq_nonneg (A*U-lam),sq_nonneg (A*U-B*T),sq_nonneg (lam-B*T)] |
| have hxm := mul_le_mul_of_nonneg_right hx hM |
| have he : (∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z psi) = |
| (∫ z in innerFieldTestAnnulus h, (max (normalizedCoreField Z lam psi z) 0)^2)*M := by |
| exact integral_mul_const M _ |
| unfold actualInnerBoundaryCap innerCapStatistic |
| rw [he] |
| change (A*U+lam+B*T)^2*M ≤ |
| 3*A^2*innerFieldVolume h*((∫ z in innerFieldTestAnnulus h, (max (normalizedCoreField Z lam psi z) 0)^2)*M)+ |
| 3*lam^2*M+3*B^2*rawWeightedMoment psi (coulombTailWeight h) |
| have h1 := mul_le_mul_of_nonneg_left hU (show 0 ≤ 3*A^2 by positivity) |
| have h2 := mul_le_mul_of_nonneg_left hT (show 0 ≤ 3*B^2 by positivity) |
| nlinarith only [hxm,h1,h2] |
|
|
| theorem actual_inner_exterior_sq_statistic {N : ℕ} {psi : FormVector N} |
| (hpsi : SobolevVector psi) (hm : formMass psi ≠ 0) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) {y : Space} (hy : 2*h ≤ ‖y‖) : |
| (max (sourceField Z (actualInnerSource psi h) y) 0)^2*formMass psi ≤ |
| innerCapStatistic psi Z lam h*((2*h)/‖y‖)^2 := by |
| have hcap := actual_inner_exterior_cap hpsi hm hZ hlam hh hy |
| have hc := actualInnerBoundaryCap_nonneg psi Z hlam hh |
| have hq : 0 ≤ (2*h)/‖y‖ := by positivity |
| have hf : max (sourceField Z (actualInnerSource psi h) y) 0 ≤ |
| actualInnerBoundaryCap psi Z lam h*((2*h)/‖y‖) := by |
| apply max_le |
| · simpa only [mul_div_assoc] using hcap |
| · exact mul_nonneg hc hq |
| have hp := mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (le_max_right _ _) hf 2) (formMass_nonneg psi) |
| calc |
| _ ≤ (actualInnerBoundaryCap psi Z lam h*((2*h)/‖y‖))^2*formMass psi := hp |
| _ = (actualInnerBoundaryCap psi Z lam h)^2*formMass psi*((2*h)/‖y‖)^2 := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right (actualInnerBoundaryCap_sq_le_statistic hpsi hZ hlam hh) (sq_nonneg _) |
|
|
| end |
| open MeasureTheory Filter Set Metric |
| open scoped BigOperators |
|
|
| def innerFieldAnnulusWeight (h : ℝ) : Space → ℝ := |
| (innerFieldTestAnnulus h).indicator (fun _ => 1) |
| lemma innerFieldAnnulusWeight_integrable (h : ℝ) : Integrable (innerFieldAnnulusWeight h) := by |
| rw [innerFieldAnnulusWeight,integrable_indicator_iff (innerFieldTestAnnulus_measurable h)] |
| exact integrableOn_const (C := (1:ℝ)) (hs := innerFieldTestAnnulus_finite h |>.ne) |
| lemma innerFieldAnnulusWeight_nonneg (h : ℝ) (z : Space) : 0 ≤ innerFieldAnnulusWeight h z := |
| indicator_nonneg (fun _ _ => zero_le_one) z |
| lemma innerFieldAnnulusWeight_separation (h : ℝ) (z : Space) |
| (hz : innerFieldAnnulusWeight h z ≠ 0) : h ≤ ‖z‖ := by |
| by_contra! hc |
| exact hz (indicator_of_notMem (by intro hd; exact (not_le_of_gt hc) hd.1) _) |
| lemma innerFieldAnnulus_weight_eq (h : ℝ) (f : Space → ℝ) : |
| (∫ z, f z*innerFieldAnnulusWeight h z) = ∫ z in innerFieldTestAnnulus h, f z := by |
| rw [←integral_indicator (innerFieldTestAnnulus_measurable h)] |
| apply integral_congr_ae |
| exact ae_of_all _ (fun z => by by_cases hz : z ∈ innerFieldTestAnnulus h <;> simp [innerFieldAnnulusWeight,hz]) |
|
|
| lemma coreFieldSquare_annular_statistic_integrable {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (t : Spins M) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| Integrable (fun u => ∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z (coreSlice ψ t u)) := by |
| have hi := (coreFieldSquare_weighted_prod_integrable hψ t hZ hlam hh |
| (innerFieldAnnulusWeight_integrable h) (innerFieldAnnulusWeight_nonneg h) |
| (innerFieldAnnulusWeight_separation h)).integral_prod_left |
| simpa only [innerFieldAnnulus_weight_eq] using hi |
|
|
| lemma coreFieldSquare_annular_fubini {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| coreLawAverage ψ (fun φ => ∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z φ) = |
| ∫ z in innerFieldTestAnnulus h, coreLawAverage ψ (coreFieldSquare Z lam z) := by |
| have he := coreFieldSquare_weighted_fubini hψ Z lam hZ hlam hh |
| (innerFieldAnnulusWeight_integrable h) (innerFieldAnnulusWeight_nonneg h) |
| (innerFieldAnnulusWeight_separation h) |
| simpa only [innerFieldAnnulus_weight_eq] using he |
|
|
| lemma innerCapStatistic_coreSlice_integrable {N M : ℕ} {ψ : FormVector (N+M)} |
| (hψ : SobolevVector ψ) (t : Spins M) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| Integrable (fun u => innerCapStatistic (coreSlice ψ t u) Z lam h) := by |
| unfold innerCapStatistic |
| exact (((coreFieldSquare_annular_statistic_integrable hψ t hZ hlam hh).const_mul _).add |
| ((hψ.coreSlice_mass_integrable t).const_mul _)).add |
| ((rawWeightedMoment_coreSlice_integrable hψ (coulombTailWeight_measurable h) |
| (coulombTailWeight_nonneg h) (coulombTailWeight_le hh) t).const_mul _) |
|
|
| namespace CoreObservationGraph |
|
|
| def innerCapAverage (G : CoreObservationGraph) (Z lam h : ℝ) : ℝ := |
| coreLawAverage G.vector (fun φ => innerCapStatistic φ Z lam h) |
|
|
| lemma fieldMoment_annular_integrable (G : CoreObservationGraph) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| IntegrableOn (G.fieldMoment Z lam) (innerFieldTestAnnulus h) := by |
| have hi := G.fieldMoment_weighted_integrable hZ hlam hh (innerFieldAnnulusWeight_integrable h) |
| (innerFieldAnnulusWeight_nonneg h) (innerFieldAnnulusWeight_separation h) |
| rw [←integrable_indicator_iff (innerFieldTestAnnulus_measurable h)] |
| apply hi.congr |
| exact ae_of_all _ (fun z => by by_cases hz : z ∈ innerFieldTestAnnulus h <;> simp [innerFieldAnnulusWeight,hz]) |
|
|
| lemma innerCapAverage_eq (G : CoreObservationGraph) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| G.innerCapAverage Z lam h = |
| 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h* |
| (∫ z in innerFieldTestAnnulus h, G.fieldMoment Z lam z)+ |
| 3*lam^2*G.mass+3*(16*packetPotentialConstant)^2*G.weightedMoment (coulombTailWeight h) := by |
| have he : G.innerCapAverage Z lam h = |
| 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h* |
| coreLawAverage G.vector (fun φ => ∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z φ)+ |
| 3*lam^2*coreLawAverage G.vector formMass+ |
| 3*(16*packetPotentialConstant)^2*G.weightedMoment (coulombTailWeight h) := by |
| unfold innerCapAverage coreLawAverage innerCapStatistic weightedMoment coreLawAverage |
| rw [Finset.mul_sum,Finset.mul_sum,Finset.mul_sum,←Finset.sum_add_distrib,←Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro t _ |
| dsimp only |
| have hA : Integrable (fun v => 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h* |
| (∫ z in innerFieldTestAnnulus h, coreFieldSquare Z lam z (coreSlice G.vector t v))) := |
| (coreFieldSquare_annular_statistic_integrable G.sobolev t hZ hlam hh).const_mul _ |
| have hB : Integrable (fun v => 3*lam^2*formMass (coreSlice G.vector t v)) := |
| (G.sobolev.coreSlice_mass_integrable t).const_mul _ |
| have hC : Integrable (fun v => 3*(16*packetPotentialConstant)^2* |
| rawWeightedMoment (coreSlice G.vector t v) (coulombTailWeight h)) := |
| (rawWeightedMoment_coreSlice_integrable G.sobolev (coulombTailWeight_measurable h) |
| (coulombTailWeight_nonneg h) (coulombTailWeight_le hh) t).const_mul _ |
| have hAB := integral_add hA hB |
| have hABC := integral_add (hA.add hB) hC |
| simp only [Pi.add_apply] at hABC hAB |
| rw [hABC,hAB] |
| simp only [integral_const_mul] |
| rw [he,coreFieldSquare_annular_fubini G.sobolev hZ hlam hh,coreLawAverage_mass G.sobolev] |
| rfl |
| end CoreObservationGraph |
|
|
| namespace CoreObservationEnsemble |
|
|
| def innerCapAverage (E : CoreObservationEnsemble) (Z lam h : ℝ) : ℝ := |
| ∑ i, (E.graph i).innerCapAverage Z lam h |
|
|
| lemma fieldMoment_annular_integrable (E : CoreObservationEnsemble) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| IntegrableOn (E.fieldMoment Z lam) (innerFieldTestAnnulus h) := |
| integrable_finsetSum _ (fun i _ => (E.graph i).fieldMoment_annular_integrable hZ hlam hh) |
|
|
| lemma innerCapAverage_eq (E : CoreObservationEnsemble) {Z lam h : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) : |
| E.innerCapAverage Z lam h = |
| 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h* |
| (∫ z in innerFieldTestAnnulus h, E.fieldMoment Z lam z)+ |
| 3*lam^2*E.mass+3*(16*packetPotentialConstant)^2*E.weightedMoment (coulombTailWeight h) := by |
| unfold innerCapAverage |
| simp_rw [CoreObservationGraph.innerCapAverage_eq _ hZ hlam hh] |
| simp only [Finset.sum_add_distrib,←Finset.mul_sum] |
| unfold fieldMoment mass weightedMoment |
| rw [integral_finsetSum _ (fun i _ => (E.graph i).fieldMoment_annular_integrable hZ hlam hh)] |
|
|
| lemma innerCapAverage_bound (E : CoreObservationEnsemble) {Z lam h F T : ℝ} |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) (hm : E.mass = 1) |
| (hf : ∀ z ∈ innerFieldTestAnnulus h, E.fieldMoment Z lam z ≤ F) |
| (ht : E.weightedMoment (coulombTailWeight h) ≤ T) : |
| E.innerCapAverage Z lam h ≤ |
| 3*(packetDensityConstant/(h/4)^3)^2*(innerFieldVolume h)^2*F+ |
| 3*lam^2+3*(16*packetPotentialConstant)^2*T := by |
| rw [innerCapAverage_eq E hZ hlam hh,hm,mul_one] |
| have hI : (∫ z in innerFieldTestAnnulus h, E.fieldMoment Z lam z) ≤ innerFieldVolume h*F := by |
| let : IsFiniteMeasure (volume.restrict (innerFieldTestAnnulus h)) := |
| ⟨by rw [Measure.restrict_apply_univ]; exact innerFieldTestAnnulus_finite h⟩ |
| have hb := integral_mono_ae (E.fieldMoment_annular_integrable hZ hlam hh) |
| (integrable_const F) (ae_restrict_of_forall_mem (innerFieldTestAnnulus_measurable h) hf) |
| simpa only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul,innerFieldVolume] using hb |
| have hcoef : 0 ≤ 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h := by |
| have := innerFieldVolume_nonneg h; positivity |
| calc |
| _ ≤ 3*(packetDensityConstant/(h/4)^3)^2*innerFieldVolume h*(innerFieldVolume h*F)+ |
| 3*lam^2+3*(16*packetPotentialConstant)^2*T := |
| add_le_add (add_le_add (mul_le_mul_of_nonneg_left hI hcoef) le_rfl) |
| (mul_le_mul_of_nonneg_left ht (by positivity)) |
| _ = _ := by ring |
| end CoreObservationEnsemble |
|
|
| end CoulombAtom |
|
|
| end |
|
|
| end OAI |
|
|
