| 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} |
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| 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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|
