openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /ConditionalObservables.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Variational.BlockSwap | |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.ConditionalFieldIdentity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set | |
| open scoped BigOperators | |
| namespace CoulombAtom | |
| open CoulombNeumann | |
| lemma weighted_coreSlice_integrable {N M : ℕ} (ψ : FormVector (N+M)) (s : Spins M) | |
| (F : Configuration M → ℝ) | |
| (hF : ∀ t : Spins N, Integrable (fun p : Configuration M × Configuration N => | |
| F p.1*‖(swapBlocks ψ).value (joinLists s t) (joinLists p.1 p.2)‖^2)) : | |
| Integrable (fun x => F x*formMass (coreSlice ψ s x)) := by | |
| have hi (t : Spins N) : Integrable (fun x : Configuration M => | |
| F x*(∫ y : Configuration N, ‖(coreSlice ψ s x).value t y‖^2)) := by | |
| simpa only [swapBlocks_value,coreSlice,integral_const_mul] using (hF t).integral_prod_left | |
| simpa only [formMass,Finset.mul_sum] using integrable_finsetSum Finset.univ (fun t _ => hi t) | |
| lemma retained_pressure_weighted_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) {b : ℝ} (hb : 0 < b) (s : Spins M) | |
| (S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) : | |
| Integrable (fun x => retainedPressure b (S x) x*formMass (coreSlice ψ s x)) := | |
| weighted_coreSlice_integrable ψ s _ (fun t => retained_pressure_join_integrable | |
| (swapBlocks_sobolev hψ) hb (fun p => S p.1) (fun i => measurable_fst (hS i)) s t) | |
| lemma retained_direct_weighted_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) {b : ℝ} (hb : 0 < b) (s : Spins M) | |
| (S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) : | |
| Integrable (fun x => retainedDirect b (S x) x*formMass (coreSlice ψ s x)) := | |
| weighted_coreSlice_integrable ψ s _ (fun t => retained_direct_join_integrable | |
| (swapBlocks_sobolev hψ) hb (fun p => S p.1) (fun i => measurable_fst (hS i)) s t) | |
| lemma conditionalFieldParticle_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) (s : Spins M) (i : Fin M) : | |
| Integrable (fun x => formMass (coreSlice ψ s x)*normalizedCoreField Z lam (coreSlice ψ s x) (x i)) := by | |
| have hh := (((coreSlice_mass_div_integrable hψ s i).const_mul Z).sub | |
| (coreSlice_cross_integrable hψ s i)).sub ((hψ.coreSlice_mass_integrable s).const_mul lam) | |
| apply hh.congr | |
| filter_upwards [hψ.ae_coreSlice s] with x hx | |
| simp only [Pi.sub_apply] | |
| rw [mul_comm (formMass _),normalizedCoreField_mul_mass hx] | |
| ring | |
| def conditionalRetainedFieldSum {N M : ℕ} (Z lam : ℝ) (ψ : FormVector (N+M)) | |
| (s : Spins M) (S : Configuration M → Finset (Fin M)) (x : Configuration M) : ℝ := | |
| formMass (coreSlice ψ s x)*∑ i ∈ S x, normalizedCoreField Z lam (coreSlice ψ s x) (x i) | |
| lemma conditionalRetainedFieldSum_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) (s : Spins M) | |
| (S : Configuration M → Finset (Fin M)) (hS : ∀ i, MeasurableSet {x | i ∈ S x}) : | |
| Integrable (conditionalRetainedFieldSum Z lam ψ s S) := by | |
| classical | |
| have hi (i : Fin M) := (conditionalFieldParticle_integrable hψ Z lam s i).indicator (hS i) | |
| have he (x : Configuration M) : conditionalRetainedFieldSum Z lam ψ s S x = | |
| ∑ i : Fin M, if i ∈ S x then | |
| formMass (coreSlice ψ s x)*normalizedCoreField Z lam (coreSlice ψ s x) (x i) else 0 := by | |
| simp only [conditionalRetainedFieldSum,Finset.mul_sum,Finset.sum_ite_mem,Finset.univ_inter] | |
| change Integrable (fun x => conditionalRetainedFieldSum Z lam ψ s S x) | |
| simp_rw [he] | |
| exact integrable_finsetSum _ (fun i _ => by | |
| convert hi i using 1 | |
| funext x | |
| simp only [Set.indicator_apply,Set.mem_ofPred_eq]) | |
| end CoulombAtom | |
| end | |
| end OAI | |