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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Hierarchy.BlockDerivativeSum
import OAI.Probability.SKBarriers.Interpolation.FiniteCovarianceComparison
namespace OAI
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
attribute [local instance 2000] parameterNormedGroup parameterNormedSpace
variable {S : Type}
def blockObservable {D N k : ℕ} (H : Fin D → S → ℝ)
(F : Fin (k+1) → Fin N → S → ℝ) : Fin (blockDimension D N k) → S → ℝ :=
Fin.addCases H (fun t => F (finProdFinEquiv.symm t).1 (finProdFinEquiv.symm t).2)
def blockInterpolationChoice (D N k : ℕ) : Fin (blockDimension D N k) → Bool :=
Fin.addCases (fun _ => true) (fun _ => false)
theorem blockObservable_disorder {D N k : ℕ} (H : Fin D → S → ℝ)
(F : Fin (k+1) → Fin N → S → ℝ) (i : Fin D) :
blockObservable H F (Fin.castAdd ((k+1)*N) i)=H i := by
simp only [blockObservable,Fin.addCases_left]
theorem blockObservable_field {D N k : ℕ} (H : Fin D → S → ℝ)
(F : Fin (k+1) → Fin N → S → ℝ) (b : Fin (k+1)) (i : Fin N) :
blockObservable H F (fieldIndex D N k b i)=F b i := by
rw [fieldIndex_eq_natAdd]
simp only [blockObservable,Fin.addCases_right,Equiv.symm_apply_apply]
theorem blockObservable_covariance {D N k : ℕ} (H : Fin D → S → ℝ)
(F : Fin (k+1) → Fin N → S → ℝ) (s t : S) :
(∑ i, interpolationRate (blockDimension D N k) (blockInterpolationChoice D N k) i*
(blockObservable H F i s*blockObservable H F i t))=
(1/2:ℝ)*(∑ i, H i s*H i t)-(1/2:ℝ)*(∑ b, ∑ i, F b i s*F b i t) := by
rw [Fin.sum_univ_add]
simp only [interpolationRate,blockInterpolationChoice,blockObservable,Fin.addCases_left,
Fin.addCases_right,Bool.false_eq_true,ite_true,ite_false]
rw [← finProdFinEquiv.sum_comp (fun u : Fin ((k+1)*N) =>
-(1/2:ℝ)*(F (finProdFinEquiv.symm u).1 (finProdFinEquiv.symm u).2 s*
F (finProdFinEquiv.symm u).1 (finProdFinEquiv.symm u).2 t))]
simp only [Equiv.symm_apply_apply,Fintype.sum_prod_type,← Finset.mul_sum]
ring
theorem blockObservable_partial_covariance {D N k : ℕ} (H : Fin D → S → ℝ)
(F : Fin (k+1) → Fin N → S → ℝ) (b : Fin (k+1)) (s t : S) :
partialObservableCovariance (blockDimension D N k) (blockObservable H F)
(interpolationRate (blockDimension D N k) (blockInterpolationChoice D N k))
(blockLevel D N k b) s t=
(1/2:ℝ)*(∑ i, H i s*H i t)-
(1/2:ℝ)*(∑ a, if a≤b then ∑ i, F a i s*F a i t else 0) := by
unfold partialObservableCovariance
rw [Fin.sum_univ_add]
have hd (i : Fin D) : (Fin.castAdd ((k+1)*N) i).val < (blockLevel D N k b).val := by
simp only [Fin.val_castAdd,blockLevel_val]
omega
simp only [hd,ite_true,interpolationRate,blockInterpolationChoice,blockObservable,
Fin.addCases_left,Fin.addCases_right,Bool.false_eq_true,ite_false]
rw [← finProdFinEquiv.sum_comp (fun u : Fin ((k+1)*N) =>
if (Fin.natAdd D u).val < (blockLevel D N k b).val then
-(1/2:ℝ)*(F (finProdFinEquiv.symm u).1 (finProdFinEquiv.symm u).2 s*
F (finProdFinEquiv.symm u).1 (finProdFinEquiv.symm u).2 t) else 0)]
simp only [Equiv.symm_apply_apply,Fintype.sum_prod_type,← fieldIndex_eq_natAdd,
fieldIndex_lt_blockLevel_iff,← Finset.mul_sum]
congr 1
simp only [Finset.sum_ite_irrel,Finset.mul_sum]
rw [← Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro a _
split_ifs <;> simp only [neg_mul,mul_zero,Finset.sum_const_zero,Finset.sum_neg_distrib,← Finset.mul_sum,neg_zero]
end SK.Analytic
end
end
end OAI
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