openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Fermionic /WeightedCoreStatistics.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Localization.FiniteCellObservation | |
| namespace OAI | |
| open MeasureTheory Filter Set | |
| open scoped BigOperators | |
| noncomputable section | |
| namespace CoulombAtom | |
| def rawWeightedCount {N : ℕ} (f : Space → ℝ) (x : Configuration N) : ℝ := ∑ i, f (x i) | |
| lemma rawWeightedCount_nonneg {N : ℕ} {f : Space → ℝ} (hf : ∀ z, 0 ≤ f z) | |
| (x : Configuration N) : 0 ≤ rawWeightedCount f x := Finset.sum_nonneg (fun i _ => hf (x i)) | |
| lemma rawWeightedCount_le {N : ℕ} {f : Space → ℝ} {A : ℝ} (hf : ∀ z, f z ≤ A) | |
| (x : Configuration N) : rawWeightedCount f x ≤ N*A := by | |
| calc _ ≤ ∑ _i : Fin N, A := Finset.sum_le_sum (fun i _ => hf (x i)) | |
| _ = _ := by simp | |
| lemma rawWeightedCount_measurable {N : ℕ} {f : Space → ℝ} (hf : Measurable f) : | |
| Measurable (rawWeightedCount (N := N) f) := | |
| Finset.measurable_sum _ (fun i _ => hf.comp (measurable_pi_apply i)) | |
| lemma rawWeightedCount_cutCore_le {N : ℕ} {f : Space → ℝ} (hf : ∀ z, 0 ≤ f z) | |
| (c : Fin N → Fin 2) (x : Configuration N) : | |
| rawWeightedCount f (cutCorePositions c x) ≤ rawWeightedCount f x := by | |
| rw [rawWeightedCount,sum_cutCorePositions c x f] | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| split_ifs | |
| · exact le_rfl | |
| · exact hf _ | |
| def rawWeightedMoment {N : ℕ} (psi : FormVector N) (f : Space → ℝ) : ℝ := | |
| rawFormPair psi (fun x => rawWeightedCount f x^2) | |
| lemma rawWeightedMoment_nonneg {N : ℕ} (psi : FormVector N) (f : Space → ℝ) : | |
| 0 ≤ rawWeightedMoment psi f := | |
| Finset.sum_nonneg (fun _ _ => integral_nonneg (fun _ => mul_nonneg (sq_nonneg _) (sq_nonneg _))) | |
| lemma rawWeightedMoment_eq_integral {N : ℕ} {psi : FormVector N} (hpsi : SobolevVector psi) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| rawWeightedMoment psi f = ∫ x, rawWeightedCount f x^2 ∂formRawLaw psi := | |
| rawFormPair_eq_integral hpsi ((rawWeightedCount_measurable hm).pow_const 2) | |
| (fun x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn x) (rawWeightedCount_le hb x)) | |
| lemma rawWeightedMoment_coreSlice_integrable {N M : ℕ} {psi : FormVector (N+M)} | |
| (hpsi : SobolevVector psi) {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) | |
| {A : ℝ} (hb : ∀ z, f z ≤ A) (t : Spins M) : | |
| Integrable (fun v => rawWeightedMoment (coreSlice psi t v) f) := | |
| rawFormPair_coreSlice_integrable hpsi ((rawWeightedCount_measurable hm).pow_const 2) | |
| (fun x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn x) (rawWeightedCount_le hb x)) t | |
| lemma sum_cutCore_weightedMoment_le {N : ℕ} {psi : FormVector N} (hpsi : SobolevVector psi) | |
| (p : Fin 2 → SmoothMultiplier spaceDirections) (hp : ∀ x, ∑ a, (p a).value x^2 = 1) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| (∑ c : Fin N → Fin 2, coreLawAverage (orderedCutForm p hp psi c) | |
| (fun phi => rawWeightedMoment phi f)) ≤ rawWeightedMoment psi f := by | |
| simp only [rawWeightedMoment] | |
| simp_rw [cutCore_rawFormPair p hp hpsi _ ((rawWeightedCount_measurable hm).pow_const 2) | |
| (fun x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn x) (rawWeightedCount_le hb x))] | |
| rw [rawFormPair_eq_integral hpsi ((rawWeightedCount_measurable hm).pow_const 2) | |
| (fun x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn x) (rawWeightedCount_le hb x))] | |
| change spatialCutExpectation p hp psi (fun c x => rawWeightedCount f (cutCorePositions c x)^2) ≤ _ | |
| apply spatialCutExpectation_le_raw p hp hpsi | |
| (fun c => ((rawWeightedCount_measurable hm).comp (cutCorePositions_measurable c)).pow_const 2) | |
| (A := ((N:ℝ)*A)^2) | |
| (fun c x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn _) | |
| ((rawWeightedCount_cutCore_le hn c x).trans (rawWeightedCount_le hb x))) | |
| ((rawWeightedCount_measurable hm).pow_const 2) (B := ((N:ℝ)*A)^2) | |
| (fun x => norm_sq_le_of_nonneg (rawWeightedCount_nonneg hn _) (rawWeightedCount_le hb _)) | |
| intro c x _ | |
| exact pow_le_pow_left₀ (rawWeightedCount_nonneg hn _) (rawWeightedCount_cutCore_le hn c x) 2 | |
| lemma nextCoreObservation_weighted_statistic_integrable {N M : ℕ} {psi : FormVector (N+M)} | |
| (hpsi : SobolevVector psi) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) (c : Fin N → Fin 2) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) (t : Spins M) : | |
| Integrable (fun v => coreLawAverage (orderedCutForm p hp (coreSlice psi t v) c) | |
| (fun phi => rawWeightedMoment phi f)) := | |
| nextCoreObservation_statistic_integrable psi p hp c (fun phi => rawWeightedMoment phi f) | |
| (fun s => rawWeightedMoment_coreSlice_integrable (nextCoreObservation_sobolev hpsi p hp c) hm hn hb s) t | |
| theorem sum_nextCoreObservation_weightedMoment_le {N M : ℕ} {psi : FormVector (N+M)} | |
| (hpsi : SobolevVector psi) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| (∑ c : Fin N → Fin 2, coreLawAverage (nextCoreObservation p hp psi c) | |
| (fun phi => rawWeightedMoment phi f)) ≤ coreLawAverage psi (fun phi => rawWeightedMoment phi f) := by | |
| have ht (c : Fin N → Fin 2) := coreLawAverage_next psi p hp c | |
| (fun phi => rawWeightedMoment phi f) | |
| (fun s => rawWeightedMoment_coreSlice_integrable (nextCoreObservation_sobolev hpsi p hp c) hm hn hb s) | |
| simp_rw [ht] | |
| rw [Finset.sum_comm] | |
| change (∑ t : Spins M, ∑ c : Fin N → Fin 2, ∫ v, | |
| coreLawAverage (orderedCutForm p hp (coreSlice psi t v) c) (fun phi => rawWeightedMoment phi f)) ≤ | |
| ∑ t : Spins M, ∫ v, rawWeightedMoment (coreSlice psi t v) f | |
| apply Finset.sum_le_sum | |
| intro t _ | |
| have hi := fun c (_ : c ∈ (Finset.univ : Finset (Fin N → Fin 2))) => | |
| nextCoreObservation_weighted_statistic_integrable hpsi p hp c hm hn hb t | |
| rw [←integral_finsetSum _ hi] | |
| apply integral_mono_ae (integrable_finsetSum _ hi) (rawWeightedMoment_coreSlice_integrable hpsi hm hn hb t) | |
| filter_upwards [hpsi.ae_coreSlice t] with v hv | |
| exact sum_cutCore_weightedMoment_le hv p hp hm hn hb | |
| namespace CoreObservationGraph | |
| def weightedMoment (G : CoreObservationGraph) (f : Space → ℝ) : ℝ := | |
| coreLawAverage G.vector (fun phi => rawWeightedMoment phi f) | |
| lemma observe_weightedMoment (G : CoreObservationGraph) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| (∑ c : Fin G.coreSize → Fin 2, (G.observe p hp c).weightedMoment f) ≤ G.weightedMoment f := | |
| sum_nextCoreObservation_weightedMoment_le G.sobolev p hp hm hn hb | |
| end CoreObservationGraph | |
| namespace CoreObservationEnsemble | |
| def weightedMoment (E : CoreObservationEnsemble) (f : Space → ℝ) : ℝ := | |
| ∑ i, (E.graph i).weightedMoment f | |
| lemma initial_weightedMoment {N : ℕ} (psi : FormVector N) (hpsi : SobolevFermion psi) (f : Space → ℝ) : | |
| (initial psi hpsi).weightedMoment f = rawWeightedMoment psi f := by | |
| change (∑ _i : Unit, coreLawAverage (M := 0) psi (fun phi => rawWeightedMoment phi f)) = _ | |
| simp [coreLawAverage_empty] | |
| lemma observe_weightedMoment (E : CoreObservationEnsemble) (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| (E.observe p hp).weightedMoment f ≤ E.weightedMoment f := by | |
| change (∑ ic : Σ i : E.Branch, Fin (E.graph i).coreSize → Fin 2, | |
| ((E.graph ic.1).observe p hp ic.2).weightedMoment f) ≤ _ | |
| rw [Fintype.sum_sigma] | |
| exact Finset.sum_le_sum (fun i _ => (E.graph i).observe_weightedMoment p hp hm hn hb) | |
| lemma observeCells_weightedMoment (E : CoreObservationEnsemble) (ys : List Space) | |
| {f : Space → ℝ} (hm : Measurable f) (hn : ∀ z, 0 ≤ f z) {A : ℝ} (hb : ∀ z, f z ≤ A) : | |
| (E.observeCells ys).weightedMoment f ≤ E.weightedMoment f := by | |
| classical | |
| induction ys generalizing E with | |
| | nil => exact le_rfl | |
| | cons y ys ih => | |
| apply (ih _).trans | |
| unfold observeCell | |
| split_ifs | |
| · exact le_rfl | |
| · exact observe_weightedMoment E _ _ hm hn hb | |
| end CoreObservationEnsemble | |
| end CoulombAtom | |
| end | |
| end OAI | |