openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /CoreFieldRegularity.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Variational.CorePriceExcess | |
| import OAI.Analysis.CoulombIonization.ThomasFermi.PatchTF | |
| import OAI.Analysis.CoulombIonization.Variational.PatchMeasurable | |
| namespace OAI | |
| noncomputable section | |
| namespace CoulombAtom | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| section Work_CoreFieldRegularity_scope | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators ContDiff | |
| lemma coreCoulombAt_nonneg {N : ℕ} (ψ : FormVector N) (z : Space) : | |
| 0 ≤ coreCoulombAt ψ z := by | |
| apply Finset.sum_nonneg | |
| intro s _ | |
| apply Finset.sum_nonneg | |
| intro i _ | |
| exact integral_nonneg (fun x => div_nonneg (sq_nonneg _) (norm_nonneg _)) | |
| lemma coreCoulombTerm_measurable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (s : Spins N) (i : Fin N) : | |
| Measurable (fun z : Space => ∫ x, ‖ψ.value s x‖^2/‖x i-z‖) := by | |
| let h := (hψ.1 s).aestronglyMeasurable | |
| let v := h.mk (ψ.value s) | |
| have hv : Measurable v := h.stronglyMeasurable_mk.measurable | |
| have hm : StronglyMeasurable (fun q : Space × Configuration N => ‖v q.2‖^2/‖q.2 i-q.1‖) := | |
| (((hv.comp measurable_snd).norm.pow_const 2).div (by fun_prop)).stronglyMeasurable | |
| have hh : Measurable (fun z : Space => ∫ x, ‖v x‖^2/‖x i-z‖) := hm.integral_prod_right.measurable | |
| convert hh using 1 | |
| funext z | |
| apply integral_congr_ae | |
| filter_upwards [h.ae_eq_mk] with x hx | |
| rw [hx] | |
| lemma coreCoulombAt_measurable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) : | |
| Measurable (coreCoulombAt ψ) := | |
| Finset.measurable_sum _ (fun s _ => Finset.measurable_sum _ (fun i _ => coreCoulombTerm_measurable hψ s i)) | |
| lemma normalizedCoreField_measurable {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (Z lam : ℝ) : Measurable (normalizedCoreField Z lam ψ) := | |
| (((measurable_const.div continuous_norm.measurable).sub | |
| (coreCoulombAt_measurable hψ)).div_const _).sub_const _ | |
| lemma coreCoulombTerm_le_separated {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (A : Set Space) (hc : ∀ x i, x i ∉ A → FormZeroAt ψ x) | |
| {d : ℝ} (hd : 0 < d) (z : Space) (hsep : ∀ a ∈ A, d ≤ ‖a-z‖) | |
| (s : Spins N) (i : Fin N) : | |
| (∫ x, ‖ψ.value s x‖^2/‖x i-z‖) ≤ (∫ x, ‖ψ.value s x‖^2)/d := by | |
| rw [←integral_div] | |
| apply integral_mono_of_nonneg | |
| · exact Eventually.of_forall fun x => div_nonneg (sq_nonneg _) (norm_nonneg _) | |
| · exact ((hψ.1 s).integrable_norm_pow (by decide)).div_const d | |
| · apply Eventually.of_forall | |
| intro x | |
| dsimp only | |
| by_cases hi : x i ∈ A | |
| · exact div_le_div_of_nonneg_left (sq_nonneg _) hd (hsep _ hi) | |
| · rw [(hc x i hi s).1] | |
| simp only [norm_zero,zero_pow (by decide : (2:ℕ) ≠ 0),zero_div,le_refl] | |
| lemma coreCoulombAt_le_separated {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (A : Set Space) (hc : ∀ x i, x i ∉ A → FormZeroAt ψ x) | |
| {d : ℝ} (hd : 0 < d) (z : Space) (hsep : ∀ a ∈ A, d ≤ ‖a-z‖) : | |
| coreCoulombAt ψ z ≤ N*formMass ψ/d := by | |
| have hh := Finset.sum_le_sum (s := Finset.univ) (fun s _ => | |
| Finset.sum_le_sum (s := Finset.univ) (fun i _ => coreCoulombTerm_le_separated hψ A hc hd z hsep s i)) | |
| simpa only [coreCoulombAt,Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul, ←mul_div_assoc,←Finset.sum_div,←Finset.mul_sum,formMass] using hh | |
| lemma normalizedCoreField_abs_le {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (A : Set Space) (hc : ∀ x i, x i ∉ A → FormZeroAt ψ x) | |
| {d : ℝ} (hd : 0 < d) (z : Space) (hsep : ∀ a ∈ A, d ≤ ‖a-z‖) | |
| {r : ℝ} (hr : 0 < r) (hz : r ≤ ‖z‖) (Z lam : ℝ) : | |
| |normalizedCoreField Z lam ψ z| ≤ |Z|/r+|lam|+N/d := by | |
| by_cases hm : formMass ψ = 0 | |
| · simp only [normalizedCoreField,hm,mul_zero,zero_div,zero_sub,div_zero,abs_neg] | |
| have : 0 ≤ (N:ℝ)/d := div_nonneg (Nat.cast_nonneg _) hd.le | |
| have : 0 ≤ |Z|/r := div_nonneg (abs_nonneg _) hr.le | |
| linarith | |
| · have hmass : 0 < formMass ψ := lt_of_le_of_ne (formMass_nonneg ψ) (Ne.symm hm) | |
| have hn := coreCoulombAt_nonneg ψ z | |
| have hu := coreCoulombAt_le_separated hψ A hc hd z hsep | |
| have he : normalizedCoreField Z lam ψ z = Z/‖z‖-coreCoulombAt ψ z/formMass ψ-lam := by | |
| unfold normalizedCoreField | |
| field_simp | |
| rw [he] | |
| have hab := abs_sub (Z/‖z‖-coreCoulombAt ψ z/formMass ψ) lam | |
| have hab' := abs_sub (Z/‖z‖) (coreCoulombAt ψ z/formMass ψ) | |
| have hZ : |Z/‖z‖| ≤ |Z|/r := by | |
| rw [abs_div,abs_of_nonneg (norm_nonneg _)] | |
| exact div_le_div_of_nonneg_left (abs_nonneg _) hr hz | |
| have hc' : |coreCoulombAt ψ z/formMass ψ| ≤ N/d := by | |
| rw [abs_of_nonneg (div_nonneg hn hmass.le)] | |
| apply (div_le_iff₀ hmass).mpr | |
| simpa only [mul_div_assoc, mul_comm (formMass ψ), div_mul_eq_mul_div] using hu | |
| linarith | |
| open CoulombAnalysis | |
| lemma normalizedCoreField_memLp_patch {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (A : Set Space) (hc : ∀ x i, x i ∉ A → FormZeroAt ψ x) | |
| (y : Space) (R : ℝ) {d r : ℝ} (hd : 0 < d) (hr : 0 < r) | |
| (hnuc : ∀ z ∈ closedBall y R, r ≤ ‖z‖) | |
| (hsep : ∀ a ∈ A, ∀ z ∈ closedBall y R, d ≤ ‖a-z‖) (Z lam : ℝ) : | |
| MemLp (fun x => normalizedCoreField Z lam ψ (y+x)) (5/2) (ballMeasure R) := by | |
| apply MemLp.of_bound | |
| (((normalizedCoreField_measurable hψ Z lam).comp (measurable_const.add measurable_id)).aestronglyMeasurable) | |
| (|Z|/r+|lam|+N/d) | |
| filter_upwards [ae_restrict_mem measurableSet_ball] with x hx | |
| have hz : y+x ∈ closedBall y R := by | |
| simpa only [mem_closedBall,dist_eq_norm,add_sub_cancel_left] using | |
| (show ‖x‖ ≤ R from (by simpa only [mem_ball,dist_eq_norm,sub_zero] using hx : ‖x‖ < R).le) | |
| rw [Real.norm_eq_abs] | |
| exact normalizedCoreField_abs_le hψ A hc hd (y+x) (fun a ha => hsep a ha _ hz) | |
| hr (hnuc _ hz) Z lam | |
| end Work_CoreFieldRegularity_scope | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators ENNReal | |
| open CoulombAnalysis | |
| lemma coreSlice_field_measurable_rep {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (t : Spins M) (Z lam : ℝ) : | |
| ∃ F : Configuration M × Space → ℝ, Measurable F ∧ | |
| ∀ᵐ u : Configuration M, ∀ z : Space, | |
| F (u,z) = normalizedCoreField Z lam (coreSlice ψ t u) z := by | |
| let d : Spins N → Configuration N × Configuration M → ℝ := | |
| fun s q => ‖ψ.value (joinLists s t) (joinLists q.1 q.2)‖^2 | |
| have hd (s : Spins N) : AEStronglyMeasurable (d s) := | |
| (integrable_join (N := N) (M := M) (hψ.1 (joinLists s t)).norm.integrable_sq).aestronglyMeasurable | |
| let v : Spins N → Configuration N × Configuration M → ℝ := fun s => (hd s).mk (d s) | |
| have hv (s : Spins N) : Measurable (v s) := (hd s).stronglyMeasurable_mk.measurable | |
| have he (s : Spins N) : ∀ᵐ u : Configuration M, ∀ᵐ x : Configuration N, | |
| d s (x,u) = v s (x,u) := by | |
| have hh : d s =ᵐ[volume] v s := (hd s).ae_eq_mk | |
| rw [Measure.volume_eq_prod] at hh | |
| exact Measure.ae_ae_of_ae_prod (Measure.measurePreserving_swap.quasiMeasurePreserving.ae hh) | |
| let m : Configuration M → ℝ := fun u => ∑ s : Spins N, ∫ x : Configuration N, v s (x,u) | |
| have hm : Measurable m := by | |
| apply Finset.measurable_sum | |
| intro s _ | |
| exact (StronglyMeasurable.integral_prod_left (f := fun x u => v s (x,u)) | |
| (hv s).stronglyMeasurable).measurable | |
| let V : Configuration M × Space → ℝ := fun q => | |
| ∑ s : Spins N, ∑ i : Fin N, ∫ x : Configuration N, v s (x,q.1)/‖x i-q.2‖ | |
| have hV : Measurable V := by | |
| apply Finset.measurable_sum | |
| intro s _ | |
| apply Finset.measurable_sum | |
| intro i _ | |
| have hh : Measurable (fun q : (Configuration M × Space) × Configuration N => | |
| v s (q.2,q.1.1)/‖q.2 i-q.1.2‖) := | |
| ((hv s).comp (measurable_snd.prodMk (measurable_fst.fst))).div (by fun_prop) | |
| exact hh.stronglyMeasurable.integral_prod_right.measurable | |
| refine ⟨fun q => (Z*m q.1/‖q.2‖-V q)/m q.1-lam, ?_, ?_⟩ | |
| · exact (((measurable_const.mul (hm.comp measurable_fst)).div | |
| (continuous_norm.measurable.comp measurable_snd)).sub hV).div | |
| (hm.comp measurable_fst) |>.sub_const lam | |
| · filter_upwards [ae_all_iff.mpr he] with u hu | |
| have hm' : m u = formMass (coreSlice ψ t u) := by | |
| apply Finset.sum_congr rfl | |
| intro s _ | |
| apply integral_congr_ae | |
| filter_upwards [hu s] with x hx | |
| exact hx.symm | |
| have hV' (z : Space) : V (u,z) = coreCoulombAt (coreSlice ψ t u) z := by | |
| apply Finset.sum_congr rfl | |
| intro s _ | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| apply integral_congr_ae | |
| filter_upwards [hu s] with x hx | |
| exact congrArg (fun a : ℝ => a/‖x i-z‖) hx.symm | |
| intro z | |
| rw [hm',hV'] | |
| rfl | |
| local instance : Fact ((5/2:ℝ≥0∞) ≠ ⊤) := ⟨by finiteness⟩ | |
| local instance (R : ℝ) : MeasurableSpace (TFField R) := borel _ | |
| local instance (R : ℝ) : BorelSpace (TFField R) := ⟨rfl⟩ | |
| local instance (R : ℝ) : MeasurableSpace (TFLp (ballMeasure R)) := borel _ | |
| local instance (R : ℝ) : BorelSpace (TFLp (ballMeasure R)) := ⟨rfl⟩ | |
| def conditionalPatchField {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M) | |
| (Z lam : ℝ) (y : Space) (R : ℝ) (u : Configuration M) : TFField R := | |
| toLpOrZero (ballMeasure R) (5/2) (fun u z => normalizedCoreField Z lam (coreSlice ψ t u) (y+z)) u | |
| lemma conditionalPatchField_aemeasurable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (t : Spins M) (Z lam : ℝ) (y : Space) (R : ℝ) : | |
| AEMeasurable (conditionalPatchField ψ t Z lam y R) := by | |
| obtain ⟨F,hF,he⟩ := coreSlice_field_measurable_rep hψ t Z lam | |
| have hm : Measurable (fun q : Configuration M × Space => F (q.1,y+q.2)) := | |
| hF.comp (measurable_fst.prodMk (measurable_const.add measurable_snd)) | |
| have hh := measurable_toLpOrZero (μ := ballMeasure R) (p := 5/2) | |
| (f := fun u z => F (u,y+z)) hm | |
| apply hh.aemeasurable.congr | |
| filter_upwards [he] with u hu | |
| apply toLpOrZero_congr (f := fun u z => F (u,y+z)) | |
| (g := fun u z => normalizedCoreField Z lam (coreSlice ψ t u) (y+z)) | |
| exact ae_of_all _ (fun z => hu (y+z)) | |
| lemma conditionalPatchMinimizer_aemeasurable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (t : Spins M) (Z lam : ℝ) (y : Space) (R : ℝ) : | |
| AEMeasurable (fun u => tfPatchMinimizer R tfKinetic tfKinetic_pos | |
| (conditionalPatchField ψ t Z lam y R u)) := | |
| (tfPatchMinimizer_measurable R tfKinetic tfKinetic_pos).comp_aemeasurable | |
| (conditionalPatchField_aemeasurable hψ t Z lam y R) | |
| end CoulombAtom | |
| end | |
| end OAI | |