| import Mathlib |
| import OAI.Analysis.CoulombIonization.Localization.ObservationLikelihood |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology BigOperators ContDiff |
| section Work_ArraySymmetry_scope |
|
|
| open MeasureTheory Set Finset |
| open scoped ENNReal NNReal BigOperators ContDiff |
|
|
| namespace CoulombObservation |
|
|
| def arrayReindex {J I : Type*} [Fintype J] [Fintype I] (e : Equiv.Perm I) : |
| (J × I → ℝ) ≃ᵐ (J × I → ℝ) := |
| MeasurableEquiv.piCongrLeft (fun _ => ℝ) ((Equiv.refl J).prodCongr e.symm) |
|
|
| @[simp] lemma arrayReindex_apply {J I : Type*} [Fintype J] [Fintype I] |
| (e : Equiv.Perm I) (u : J × I → ℝ) (q : J × I) : |
| arrayReindex e u q = u (q.1,e q.2) := by |
| rcases q with ⟨j,i⟩ |
| simp [arrayReindex, MeasurableEquiv.coe_piCongrLeft, Equiv.piCongrLeft] |
|
|
| lemma arrayReindex_preserving {J I : Type*} [Fintype J] [Fintype I] |
| (e : Equiv.Perm I) : |
| MeasurePreserving (arrayReindex (J := J) e) |
| (Measure.pi (fun _ : J × I => compactNoiseLaw)) |
| (Measure.pi (fun _ : J × I => compactNoiseLaw)) := |
| measurePreserving_piCongrLeft (fun _ => compactNoiseLaw) _ |
|
|
| lemma arrayLikelihood_reindex {J I : Type*} [Fintype J] [Fintype I] |
| (b : J → ℝ) {s : Set (J × I → ℝ)} (hs : MeasurableSet s) |
| (e : Equiv.Perm I) (hinv : ∀ u, arrayReindex (J := J) e u ∈ s ↔ u ∈ s) |
| (x : I → ℝ) : arrayLikelihood b s (x ∘ e) = arrayLikelihood b s x := by |
| rw [arrayLikelihood_eq_probability b hs, arrayLikelihood_eq_probability b hs] |
| unfold Measure.real |
| congr 1 |
| have hset : (arrayReindex (J := J) e) ⁻¹' |
| shiftedObservationEvent s (observationPlacement b (x ∘ e)) = |
| shiftedObservationEvent s (observationPlacement b x) := by |
| ext u |
| change observationPlacement b (x ∘ e) + arrayReindex e u ∈ s ↔ |
| observationPlacement b x + u ∈ s |
| have he : observationPlacement b (x ∘ e) + arrayReindex e u = |
| arrayReindex e (observationPlacement b x + u) := by |
| ext q |
| simp only [Pi.add_apply, observationPlacement_apply, Function.comp_apply, arrayReindex_apply] |
| rw [he] |
| exact hinv _ |
| rw [← (arrayReindex_preserving e).measure_preimage |
| (shiftedObservationEvent_measurable hs _).nullMeasurableSet, hset] |
| |
| end CoulombObservation |
| |
| end Work_ArraySymmetry_scope |
| |
| open MeasureTheory Set Finset |
| open scoped ENNReal NNReal BigOperators |
| |
| namespace CoulombAtom |
| open CoulombObservation |
| |
| def flattenConfiguration (N : ℕ) : Configuration N →L[ℝ] (Fin N × Fin 3 → ℝ) := |
| ContinuousLinearMap.pi (fun q => (EuclideanSpace.proj q.2).comp |
| (ContinuousLinearMap.proj q.1)) |
| |
| @[simp] lemma flattenConfiguration_apply {N : ℕ} (x : Configuration N) |
| (q : Fin N × Fin 3) : flattenConfiguration N x q = x q.1 q.2 := rfl |
| |
| @[simp] lemma flattenConfiguration_direction {N : ℕ} (i : Fin N) (a : Fin 3) : |
| flattenConfiguration N (direction i a) = Pi.single (i,a) 1 := by |
| ext ⟨j,c⟩ |
| simp only [flattenConfiguration_apply, direction, Pi.single_apply, |
| Prod.mk.injEq] |
| split_ifs <;> simp_all |
| |
| lemma lineDeriv_flattenConfiguration {N : ℕ} (f : (Fin N × Fin 3 → ℝ) → ℝ) |
| (x : Configuration N) (i : Fin N) (a : Fin 3) : |
| lineDeriv ℝ (fun y => f (flattenConfiguration N y)) x (direction i a) = |
| lineDeriv ℝ f (flattenConfiguration N x) (Pi.single (i,a) 1) := by |
| unfold lineDeriv |
| simp only [map_add, map_smul, flattenConfiguration_direction] |
| |
| def flatRawLaw {N : ℕ} (F : fermionGraph N) : Measure (Fin N × Fin 3 → ℝ) := |
| Measure.map (flattenConfiguration N) (graphRawLaw F) |
| |
| lemma flatRawLaw_probability {N : ℕ} (F : fermionGraph N) |
| (hn : ‖fermionGraphValue N F‖^2 = 1) : IsProbabilityMeasure (flatRawLaw F) := by |
| let := graphRawLaw_probability F hn |
| unfold flatRawLaw |
| infer_instance |
| |
| def quantumEventLikelihood {N : ℕ} {J : Type*} [Fintype J] |
| (b : J → ℝ) (s : Set (J × (Fin N × Fin 3) → ℝ)) (x : Configuration N) : ℝ := |
| arrayLikelihood b s (flattenConfiguration N x) |
| |
| def quantumEventProbability {N : ℕ} {J : Type*} [Fintype J] |
| (F : fermionGraph N) (b : J → ℝ) (s : Set (J × (Fin N × Fin 3) → ℝ)) : ℝ := |
| ∫ x, quantumEventLikelihood b s x ∂graphRawLaw F |
| |
| lemma quantumEventProbability_eq_flat {N : ℕ} {J : Type*} [Fintype J] |
| (F : fermionGraph N) (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} |
| (hs : MeasurableSet s) : |
| quantumEventProbability F b s = ∫ x, arrayLikelihood b s x ∂flatRawLaw F := by |
| symm |
| exact integral_map (flattenConfiguration N).measurable.aemeasurable |
| ((arrayLikelihood_contDiff b hs (n := 1)).continuous.measurable.aestronglyMeasurable) |
| |
| lemma quantumEventLikelihood_nonneg {N : ℕ} {J : Type*} [Fintype J] |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) |
| (x : Configuration N) : 0 ≤ quantumEventLikelihood b s x := |
| arrayLikelihood_nonneg b hs _ |
| |
| lemma quantumEventLikelihood_le_one {N : ℕ} {J : Type*} [Fintype J] |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) |
| (x : Configuration N) : quantumEventLikelihood b s x ≤ 1 := |
| arrayLikelihood_le_one b hs _ |
| |
| lemma quantumEventLikelihood_measurable {N : ℕ} {J : Type*} [Fintype J] |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) : |
| Measurable (quantumEventLikelihood b s) := |
| (arrayLikelihood_contDiff b hs (n := 1)).continuous.measurable.comp |
| (flattenConfiguration N).measurable |
| |
| def QuantumEventSymmetric {N : ℕ} {J : Type*} [Fintype J] |
| (s : Set (J × (Fin N × Fin 3) → ℝ)) : Prop := |
| ∀ (π : Equiv.Perm (Fin N)) u, |
| arrayReindex (J := J) (π.prodCongr (Equiv.refl (Fin 3))) u ∈ s ↔ u ∈ s |
| |
| lemma quantumEventLikelihood_symmetric {N : ℕ} {J : Type*} [Fintype J] |
| (b : J → ℝ) {s : Set (J × (Fin N × Fin 3) → ℝ)} (hs : MeasurableSet s) |
| (hsy : QuantumEventSymmetric s) (π : Equiv.Perm (Fin N)) (x : Configuration N) : |
| quantumEventLikelihood b s (x ∘ π) = quantumEventLikelihood b s x := by |
| have he : flattenConfiguration N (x ∘ π) = |
| flattenConfiguration N x ∘ (π.prodCongr (Equiv.refl (Fin 3))) := by |
| ext ⟨i,a⟩ |
| rfl |
| unfold quantumEventLikelihood |
| rw [he] |
| exact arrayLikelihood_reindex b hs _ (hsy π) _ |
| |
| end CoulombAtom |
| |
| end |
| |
| end OAI |
| |
