openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /ConditionalShellField.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.ShellMeanStatistic | |
| import OAI.Analysis.CoulombIonization.Variational.FullSliceStatistics | |
| import OAI.Analysis.CoulombIonization.FormDomain.InnerLinearCap | |
| namespace OAI | |
| noncomputable section | |
| namespace CoulombAtom | |
| section | |
| open MeasureTheory Set Filter | |
| open scoped BigOperators | |
| attribute [local irreducible] oneBodySquareTotal graphFormVector fermionGraph | |
| lemma shell_closed_inner_term_integrable {N : ℕ} {psi : FormVector N} | |
| (hpsi : SobolevVector psi) (p : SmoothMultiplier spaceDirections) (Z : ℝ) | |
| {h a : ℝ} (ha : 0 < a) (hh : h ≤ a) (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖) | |
| (y : Space) (S : ℝ) (s : Spins N) : | |
| Integrable (fun x => (p.value y^2/S)*(closedRawInnerField Z h x y*‖psi.value s x‖^2)) := by | |
| by_cases hz : p.value y = 0 | |
| · simp only [hz,zero_pow (by decide : 2 ≠ 0),zero_div,zero_mul] | |
| exact integrable_zero _ _ _ | |
| have hf : Measurable (fun x : Configuration N => closedRawInnerField Z h x y) := | |
| by simpa only [Function.comp_def,id_eq] using | |
| (closedRawInnerField_measurable N Z h).comp (measurable_id.prodMk measurable_const) | |
| exact (rawFormPair_integrable hpsi hf | |
| (closedRawInnerField_bound Z ha (by linarith) (hp y hz)) s).const_mul _ | |
| lemma fullSlicePair_shellMean {N M : ℕ} {psi : FormVector (N+M)} | |
| (p : SmoothMultiplier spaceDirections) (Z : ℝ) {h a : ℝ} | |
| (ha : 0 < a) (hh : h ≤ a) (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖) | |
| (hs : SplitPositionSupport psi (fun z => p.value z = 0) (fun z => h < ‖z‖)) | |
| (t : Spins M) (v : Configuration M) (hpsi : SobolevVector (coreSlice psi t v)) | |
| (hm : formMass (coreSlice psi t v) ≠ 0) : | |
| fullSlicePair psi (shellMeanStatistic p Z ha hh hp) t v = | |
| ∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)* | |
| rawFormPair (coreSlice psi t v) (fun x => closedRawInnerField Z h x (v i)) := by | |
| have hv := hs.outer_of_mass_ne_zero t v hm | |
| have he (s : Spins N) : | |
| (∫ x : Configuration N, shellMeanField p Z h (joinLists x v)*‖(coreSlice psi t v).value s x‖^2) = | |
| ∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)* | |
| (∫ x : Configuration N, closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) := by | |
| calc | |
| _ = ∫ x : Configuration N, ∑ i : Fin M, | |
| (p.value (v i)^2/(oneBodySquareTotal p).value v)* | |
| (closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) := by | |
| apply integral_congr_ae | |
| apply ae_of_all | |
| intro x | |
| dsimp only | |
| by_cases hz : (coreSlice psi t v).value s x = 0 | |
| · simp only [hz,norm_zero,zero_pow (by decide : 2 ≠ 0),mul_zero,Finset.sum_const_zero] | |
| · rw [shellMeanField_join p Z h x v (hs s t x v hz).1 hv] | |
| simp only [Finset.sum_div,Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| ring | |
| _ = ∑ i : Fin M, ∫ x : Configuration N, | |
| (p.value (v i)^2/(oneBodySquareTotal p).value v)* | |
| (closedRawInnerField Z h x (v i)*‖(coreSlice psi t v).value s x‖^2) := | |
| integral_finsetSum _ (fun i _ => shell_closed_inner_term_integrable hpsi p Z ha hh hp _ _ s) | |
| _ = _ := by simp only [integral_const_mul] | |
| change (∑ s : Spins N, ∫ x : Configuration N, | |
| shellMeanField p Z h (joinLists x v)*‖(coreSlice psi t v).value s x‖^2) = _ | |
| simp_rw [he] | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [←Finset.mul_sum] | |
| rfl | |
| lemma fullSlicePair_shellMean_le {N M : ℕ} {psi : FormVector (N+M)} | |
| (p : SmoothMultiplier spaceDirections) {Z lam h a : ℝ} | |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) (ha : h ≤ a) | |
| (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖) | |
| (hs : SplitPositionSupport psi (fun z => p.value z = 0) (fun z => h < ‖z‖)) | |
| (t : Spins M) (v : Configuration M) (hpsi : SobolevVector (coreSlice psi t v)) : | |
| fullSlicePair psi (shellMeanStatistic p Z (hh.trans_le ha) ha hp) t v ≤ | |
| (h/a)*Real.sqrt (innerCapStatistic (coreSlice psi t v) Z lam h)* | |
| Real.sqrt (formMass (coreSlice psi t v)) := by | |
| let C := (h/a)*Real.sqrt (innerCapStatistic (coreSlice psi t v) Z lam h)* | |
| Real.sqrt (formMass (coreSlice psi t v)) | |
| have hap : 0 < a := hh.trans_le ha | |
| have hC : 0 ≤ C := by dsimp [C]; positivity | |
| by_cases hm : formMass (coreSlice psi t v) = 0 | |
| · rw [fullSlicePair_eq_zero_of_mass _ t v hpsi hm] | |
| exact hC | |
| rw [fullSlicePair_shellMean p Z (hh.trans_le ha) ha hp hs t v hpsi hm] | |
| have hS := oneBodySquareTotal_nonneg p v | |
| calc | |
| _ ≤ ∑ i : Fin M, (p.value (v i)^2/(oneBodySquareTotal p).value v)*C := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| by_cases hi : p.value (v i) = 0 | |
| · simp only [hi,zero_pow (by decide : 2 ≠ 0),zero_div,zero_mul,le_refl] | |
| · have hy := hp _ hi | |
| have hr := rawFormPair_closedRawInnerField (Z := Z) hpsi (hh.trans_le ha) | |
| (show h+a ≤ ‖v i‖ by linarith) hm | |
| rw [hr] | |
| exact mul_le_mul_of_nonneg_left (actual_inner_linear_statistic hpsi hZ hlam hh ha hy) | |
| (div_nonneg (sq_nonneg _) hS) | |
| _ = ((oneBodySquareTotal p).value v/(oneBodySquareTotal p).value v)*C := by | |
| rw [←Finset.sum_mul,←Finset.sum_div,oneBodySquareTotal_value] | |
| _ ≤ C := by | |
| by_cases hz : (oneBodySquareTotal p).value v = 0 | |
| · rw [hz,div_zero,zero_mul]; exact hC | |
| · rw [div_self hz,one_mul] | |
| end | |
| open MeasureTheory Filter | |
| open scoped BigOperators | |
| namespace CoreObservationEnsemble | |
| lemma fullAverage_shellMean_le (E : CoreObservationEnsemble) | |
| (p : SmoothMultiplier spaceDirections) {Z lam h a : ℝ} | |
| (hZ : 0 ≤ Z) (hlam : 0 ≤ lam) (hh : 0 < h) (ha : h ≤ a) | |
| (hp : ∀ y, p.value y ≠ 0 → 2*a ≤ ‖y‖) | |
| (hs : E.supported (fun z => p.value z = 0) (fun z => h < ‖z‖)) : | |
| E.fullAverage (shellMeanStatistic p Z (hh.trans_le ha) ha hp) ≤ | |
| (h/a)*Real.sqrt (E.innerCapAverage Z lam h)*Real.sqrt E.mass := by | |
| let I := Σ i : E.Branch, Spins (E.graph i).outSize | |
| let X : I → Type := fun i => Configuration (E.graph i.1).outSize | |
| let H : ∀ i : I, X i → ℝ := fun i v => innerCapStatistic (coreSlice (E.graph i.1).vector i.2 v) Z lam h | |
| let W : ∀ i : I, X i → ℝ := fun i v => formMass (coreSlice (E.graph i.1).vector i.2 v) | |
| have hH (i : I) : Integrable (H i) := innerCapStatistic_coreSlice_integrable (E.graph i.1).sobolev i.2 hZ hlam hh | |
| have hW (i : I) : Integrable (W i) := (E.graph i.1).sobolev.coreSlice_mass_integrable i.2 | |
| have hnH (i : I) (v : X i) : 0 ≤ H i v := innerCapStatistic_nonneg .. | |
| have hnW (i : I) (v : X i) : 0 ≤ W i v := formMass_nonneg _ | |
| have hc := sum_integral_sqrt_mul_sqrt_le (fun i : I => (volume : Measure (X i))) H W hH hW hnH hnW | |
| let F := shellMeanStatistic p Z (hh.trans_le ha) ha hp | |
| have hi (i : I) : | |
| (∫ v : X i, fullSlicePair (E.graph i.1).vector F i.2 v) ≤ | |
| (h/a)*(∫ v : X i, Real.sqrt (H i v)*Real.sqrt (W i v)) := by | |
| rw [←integral_const_mul] | |
| apply integral_mono_ae (fullSlicePair_integrable (E.graph i.1).sobolev F i.2) | |
| ((integrable_sqrt_mul_sqrt (hH i) (hW i) (hnH i) (hnW i)).const_mul _) | |
| filter_upwards [(E.graph i.1).sobolev.ae_coreSlice i.2] with v hv | |
| simpa only [mul_assoc] using fullSlicePair_shellMean_le p hZ hlam hh ha hp (hs i.1) i.2 v hv | |
| have hsum := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hi i) | |
| rw [←Finset.mul_sum] at hsum | |
| have heF : (∑ i : I, ∫ v : X i, fullSlicePair (E.graph i.1).vector F i.2 v) = E.fullAverage F := by | |
| rw [Fintype.sum_sigma] | |
| exact Finset.sum_congr rfl (fun i _ => fullSlicePair_fubini (E.graph i).sobolev F) | |
| have heH : (∑ i : I, ∫ v : X i, H i v) = E.innerCapAverage Z lam h := by | |
| rw [Fintype.sum_sigma] | |
| rfl | |
| have heW : (∑ i : I, ∫ v : X i, W i v) = E.mass := by | |
| rw [Fintype.sum_sigma] | |
| exact Finset.sum_congr rfl (fun i _ => (E.graph i).sobolev.integral_coreSlice_mass) | |
| rw [heF] at hsum | |
| rw [heH,heW] at hc | |
| exact hsum.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hc (div_nonneg hh.le (hh.trans_le ha).le)) | |
| end CoreObservationEnsemble | |
| theorem exists_actual_shellMean_constant : ∃ C : ℝ, 1 ≤ C ∧ | |
| ∀ (Z lam : ℝ) (N : ℕ) (psi : FormVector N), SobolevFermion psi → | |
| formMass psi = 1 → 0 ≤ Z → 0 < lam → ∀ D h u : ℝ, 0 ≤ D → 0 < h → h < u/4 → | |
| max (corePriceExcess Z lam psi) 0 ≤ D → | |
| ∀ p : SmoothMultiplier spaceDirections, | |
| (∀ y, p.value y ≠ 0 → u/2 ≤ ‖y‖ ∧ ‖y‖ ≤ 4*u) → | |
| rawFormPair psi (shellMeanField p Z h) ≤ | |
| (4*h/u)*Real.sqrt (C*(innerScale D h+lam^2)) := by | |
| obtain ⟨L,hL⟩ := exists_scaled_shell_stencil | |
| obtain ⟨C,hC,hbound⟩ := exists_actual_observed_innerCap_constant (by norm_num : (0:ℝ)<1/2) (beta := 4) L | |
| refine ⟨C,hC,?_⟩ | |
| intro Z lam N psi hpsi hm hZ hlam D h u hD hh hu he p hp | |
| have hup : 0 < u := by linarith | |
| obtain ⟨ys,hlen,hys,hcover⟩ := hL u hup | |
| let E := (CoreObservationEnsemble.initial psi hpsi).observeCells ys | |
| have hs := CoreObservationEnsemble.annular_stencil_support psi hpsi hup ys (fun y hy => (hys y hy).1) hcover | |
| have hsupport : E.supported (fun z => p.value z = 0) (fun z => h < ‖z‖) := by | |
| intro i | |
| exact (hs i).mono (fun z hz => by by_contra hn; exact hz (hp z hn)) (fun z hz => hu.trans_le hz) | |
| have hap : 0 < u/4 := by positivity | |
| have hha : h ≤ u/4 := hu.le | |
| have hpa : ∀ y, p.value y ≠ 0 → 2*(u/4) ≤ ‖y‖ := fun y hy => by linarith [(hp y hy).1] | |
| have hmean := E.fullAverage_shellMean_le p hZ hlam.le hh hha hpa hsupport | |
| have hmass : E.mass = 1 := (CoreObservationEnsemble.observeCells_mass ..).trans ((CoreObservationEnsemble.initial_mass ..).trans hm) | |
| have hav : E.innerCapAverage Z lam h ≤ C*(innerScale D h+lam^2) := | |
| hbound Z lam N psi hpsi hm hZ hlam D h u hD hh (by linarith) he ys hlen (by | |
| intro y hy; constructor <;> nlinarith [(hys y hy).1,(hys y hy).2]) | |
| have heq : E.fullAverage (shellMeanStatistic p Z hap hha hpa) = rawFormPair psi (shellMeanField p Z h) := by | |
| rw [CoreObservationEnsemble.observeCells_fullAverage,CoreObservationEnsemble.initial_fullAverage] | |
| rfl | |
| rw [heq,hmass,Real.sqrt_one,mul_one] at hmean | |
| calc | |
| _ ≤ (h/(u/4))*Real.sqrt (E.innerCapAverage Z lam h) := hmean | |
| _ ≤ (h/(u/4))*Real.sqrt (C*(innerScale D h+lam^2)) := | |
| mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hav) (div_nonneg hh.le hap.le) | |
| _ = _ := by congr 1; field_simp | |
| end CoulombAtom | |
| end | |
| end OAI | |