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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Analysis.FreeEntropy.MatrixHeatLawMoments
namespace OAI
universe u1
noncomputable section
open MeasureTheory Finset Filter
open scoped Topology
namespace FiniteEntropySeparation
theorem complex_integrable_re {Ω : Type u1} [MeasurableSpace Ω] {μ : Measure Ω} {f : Ω → ℂ}
(hf : Integrable f μ) : Integrable (fun x => (f x).re) μ := hf.re
variable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (τ : TracialVector M) {n : ℕ} (X : Fin n → H →L[ℂ] H) (hX : ∀i,IsSelfAdjoint (X i))
variable {D : ℕ → ℕ} (ν : (k:ℕ) → Measure (MatrixCoordinates n (D k)))
(hi : ∀k w,Integrable (fun a => matrixMoment a w) (ν k))
(hip : ∀k v w,Integrable (fun a => matrixMoment a v*matrixMoment a w) (ν k))
(ht : ∀w,Tendsto (fun k => ∫a,matrixMoment a w ∂ν k) atTop (𝓝 (wordMoment τ X w)))
(htp : ∀v w,Tendsto (fun k => ∫a,matrixMoment a v*matrixMoment a w ∂ν k) atTop (𝓝 (wordMoment τ X v*wordMoment τ X w)))
include hi in
theorem matrixPolynomialNorm_integrable (k : ℕ) (p : WordPolynomial n) :
Integrable (matrixPolynomialNorm p) (ν k) :=
(integrable_finsetSum p.support (fun u _ => integrable_finsetSum p.support (fun v _ => (hi k (u.reverse++v)).const_mul (star (p u)*p v)))).re
include hip in
theorem matrixPolynomialDerivative_integrable (k : ℕ) (p : WordPolynomial n) (i : Fin n) :
Integrable (matrixPolynomialDerivative p i) (ν k) := by
apply integrable_finsetSum p.support
intro w _
apply Integrable.const_mul
apply integrable_finsetSum univ
intro r _
by_cases h : w[r.val]=i
· simpa only [h,ite_true] using hip k (w.take r.val) (w.drop (r.val+1))
· simp only [h,ite_false]
exact integrable_zero _ _ _
include hi ht hX in
theorem matrixPolynomialNorm_integral_tendsto (p : WordPolynomial n) :
Tendsto (fun k => ∫a,matrixPolynomialNorm p a ∂ν k) atTop (𝓝 (‖polynomialVector τ X p‖^2)) := by
have he k : (∫a,matrixPolynomialNorm p a ∂ν k) =
(∑u∈p.support,∑v∈p.support,star (p u)*p v*(∫a,matrixMoment a (u.reverse++v) ∂ν k)).re := by
simp only [matrixPolynomialNorm]
rw [← complex_integral_re (integrable_finsetSum _ (fun u _ => integrable_finsetSum _ (fun v _ => (hi k _).const_mul _)))]
rw [integral_finsetSum _ (fun u _ => integrable_finsetSum _ (fun v _ => (hi k _).const_mul _))]
simp_rw [integral_finsetSum _ (fun v _ => (hi k _).const_mul _),integral_const_mul]
simp_rw [he]
rw [polynomialVector_norm_formula τ X hX]
exact Complex.continuous_re.continuousAt.tendsto.comp
(tendsto_finsetSum p.support (fun u _ => tendsto_finsetSum p.support (fun v _ => (ht _).const_mul _)))
include hip htp in
theorem matrixPolynomialDerivative_integral_tendsto (p : WordPolynomial n) (i : Fin n) :
Tendsto (fun k => ∫a,matrixPolynomialDerivative p i a ∂ν k) atTop (𝓝 (polynomialDerivative τ X i p)) := by
have hir (k : ℕ) (w : List (Fin n)) (r : Fin w.length) : Integrable (fun a : MatrixCoordinates n (D k) => if w[r.val]=i then matrixMoment a (w.take r.val)*matrixMoment a (w.drop (r.val+1)) else 0) (ν k) := by
by_cases h : w[r.val]=i
· simpa only [h,ite_true] using hip k (w.take r.val) (w.drop (r.val+1))
· simp only [h,ite_false]; exact integrable_zero _ _ _
have he k : (∫a,matrixPolynomialDerivative p i a ∂ν k) =
∑w∈p.support,p w*∑r : Fin w.length,if w[r.val]=i then (∫a,matrixMoment a (w.take r.val)*matrixMoment a (w.drop (r.val+1)) ∂ν k) else 0 := by
simp only [matrixPolynomialDerivative]
rw [integral_finsetSum _ (fun w _ => (integrable_finsetSum _ (fun r _ => hir k w r)).const_mul _)]
apply sum_congr rfl
intro w _
rw [integral_const_mul,integral_finsetSum _ (fun r _ => hir k w r)]
congr 1
apply sum_congr rfl
intro r _
split_ifs <;> simp
simp_rw [he]
simp only [polynomialDerivative,Finsupp.linearCombination_apply,Finsupp.sum,smul_eq_mul,derivativeMoment]
apply tendsto_finsetSum p.support
intro w _
apply Tendsto.const_mul
apply tendsto_finsetSum univ
intro r _
split_ifs
· exact htp _ _
· exact tendsto_const_nhds
include hi hip ht htp hX in
theorem matrixFisherTest_integral_tendsto (p : WordPolynomial n) (i : Fin n) :
Tendsto (fun k => ∫a,matrixFisherTest p i a ∂ν k) atTop (𝓝 (fisherTest τ X i p)) := by
have he k : (∫a,matrixFisherTest p i a ∂ν k) =
2*(∫a,matrixPolynomialDerivative p i a ∂ν k).re-∫a,matrixPolynomialNorm p a ∂ν k := by
simp only [matrixFisherTest]
rw [integral_sub ((complex_integrable_re (matrixPolynomialDerivative_integrable ν hip k p i)).const_mul 2) (matrixPolynomialNorm_integrable ν hi k p),integral_const_mul,← complex_integral_re (matrixPolynomialDerivative_integrable ν hip k p i)]
simp_rw [he]
exact ((Complex.continuous_re.continuousAt.tendsto.comp (matrixPolynomialDerivative_integral_tendsto τ X ν hip htp p i)).const_mul 2).sub
(matrixPolynomialNorm_integral_tendsto τ X hX ν hi ht p)
end FiniteEntropySeparation
end
end OAI
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