| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleScaleStats |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedStats |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
| namespace SK.Analytic |
|
|
| theorem narrowCommonSchedule_scale (β δ : ℝ) (c : List (ℝ × (ℝ × ℝ))) : |
| scaleIncrementChain β (narrowCommonSchedule δ c)=narrowCommonSchedule δ (scaleIncrementChain β c) := by |
| simp only [scaleIncrementChain,narrowCommonSchedule,List.map_map] |
| apply List.map_congr_left |
| intro p hp |
| apply Prod.ext |
| · rfl |
| · ext i |
|
|
| theorem narrowMiddleSchedule_scale (β δ : ℝ) (l : List NarrowMiddleIncrement) : |
| scaleIncrementChain β (narrowMiddleSchedule δ l)= |
| narrowMiddleSchedule δ (l.map (mapProductIncrement (fun x : ℝ × ℝ => β • x) (fun x : ℝ × ℝ => β • x))) := by |
| simp only [scaleIncrementChain,narrowMiddleSchedule,List.map_map] |
| apply List.map_congr_left |
| intro p hp |
| apply Prod.ext |
| · simp only [Function.comp_def,productMass_map] |
| · cases p < |
| norm_num [Function.comp_def,mapProductIncrement,narrowMiddleVector,smul_eq_mul] < |
|
|
| theorem narrowSchedule_scale (β δ : ℝ) (c v w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) : |
| scaleIncrementChain β (narrowSchedule δ c v w t)= |
| narrowSchedule δ (scaleIncrementChain β c) (scaleIncrementChain β v) |
| (scaleIncrementChain β w) (scaleIncrementChain β t) := by |
| simp only [narrowSchedule,scaleIncrementChain_append,narrowCommonSchedule_scale,narrowMiddleSchedule_scale, |
| tripleTailSchedule_scale,mergedProductBranches_map] |
| rfl |
|
|
| theorem narrowVariance_scale (β : ℝ) (c v w : List (ℝ × (ℝ × ℝ))) : |
| narrowVariance (scaleIncrementChain β c) (scaleIncrementChain β v) (scaleIncrementChain β w)=β^2*narrowVariance c v w := by |
| simp only [narrowVariance,weightedVariance_scale] |
| ring |
|
|
| theorem narrowAbsCross_scale (β : ℝ) (c v w : List (ℝ × (ℝ × ℝ))) : |
| narrowAbsCross (scaleIncrementChain β c) (scaleIncrementChain β v) (scaleIncrementChain β w)=β^2*narrowAbsCross c v w := by |
| simp only [narrowAbsCross,weightedAbsCross_scale] |
| ring |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
