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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
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