File size: 2,294 Bytes
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 | 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; fin_cases i <;> norm_num [Function.comp_def,smul_eq_mul] <;> ring
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 <;> ext i <;> fin_cases i <;>
norm_num [Function.comp_def,mapProductIncrement,narrowMiddleVector,smul_eq_mul] <;> ring
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
|