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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
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