| import Mathlib |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedExpansion |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem narrowRetainedCommon_variance (c : List (ℝ × (ℝ × ℝ))) : |
| ((narrowRetainedCommon c).map (fun p => (p.2.1.2+p.2.2.2)^2)).sum=weightedVariance c := by |
| simp only [narrowRetainedCommon,List.map_map,Function.comp_def,zero_add,weightedVariance] |
|
|
| theorem narrowRetainedCommon_score (c : List (ℝ × (ℝ × ℝ))) : |
| ((narrowRetainedCommon c).map (fun p => |p.2.1.2+p.2.2.2| * (|p.2.1.1|+2*|p.2.2.1|))).sum= |
| 3*weightedAbsCross c := by |
| simp only [narrowRetainedCommon,List.map_map,Function.comp_def,zero_add,weightedAbsCross] |
| rw [← List.sum_map_mul_left] |
| congr 1 |
| apply List.map_congr_left |
| intro p _ |
| ring |
|
|
| theorem narrowRetainedMiddle_variance (l : List NarrowMiddleIncrement) : |
| ((narrowRetainedMiddle l).map (fun p => (p.2.1.2+p.2.2.2)^2)).sum= |
| weightedVariance (productLeft l)+weightedVariance (productRight l) := by |
| induction l with |
| | nil => simp [narrowRetainedMiddle,productLeft,productRight,weightedVariance] |
| | cons p l ih => |
| cases p <;> simp only [narrowRetainedMiddle,List.map_cons,List.map_map,Function.comp_def,List.sum_cons, |
| productVector,Prod.snd_zero,productLeft,productRight,List.filterMap_cons,weightedVariance,List.map_cons,List.sum_cons, |
| add_zero,zero_add] at ih ⊢ <;> rw [ih] <;> ring |
|
|
| theorem narrowRetainedMiddle_score (l : List NarrowMiddleIncrement) : |
| ((narrowRetainedMiddle l).map (fun p => |p.2.1.2+p.2.2.2| * (|p.2.1.1|+2*|p.2.2.1|))).sum= |
| weightedAbsCross (productLeft l)+2*weightedAbsCross (productRight l) := by |
| induction l with |
| | nil => simp [narrowRetainedMiddle,productLeft,productRight,weightedAbsCross] |
| | cons p l ih => |
| cases p <;> simp only [narrowRetainedMiddle,List.map_cons,List.map_map,Function.comp_def,List.sum_cons, |
| productVector,Prod.fst_zero,Prod.snd_zero,productLeft,productRight,List.filterMap_cons,weightedAbsCross,List.map_cons,List.sum_cons, |
| abs_zero,mul_zero,add_zero,zero_add] at ih ⊢ <;> rw [ih] <;> ring |
|
|
| def narrowVariance (c v w : List (ℝ × (ℝ × ℝ))) := weightedVariance c+weightedVariance v+weightedVariance w |
| def narrowAbsCross (c v w : List (ℝ × (ℝ × ℝ))) := 3*weightedAbsCross c+weightedAbsCross v+2*weightedAbsCross w |
|
|
| theorem narrowRetainedPrefix_variance (c v w : List (ℝ × (ℝ × ℝ))) : |
| ((narrowRetainedPrefix c v w).map (fun p => (p.2.1.2+p.2.2.2)^2)).sum=narrowVariance c v w := by |
| simp only [narrowRetainedPrefix,List.map_append,List.sum_append,narrowRetainedCommon_variance, |
| narrowRetainedMiddle_variance,mergedProductBranches_left,mergedProductBranches_right,narrowVariance,add_assoc] |
|
|
| theorem narrowRetainedPrefix_score (c v w : List (ℝ × (ℝ × ℝ))) : |
| ((narrowRetainedPrefix c v w).map (fun p => |p.2.1.2+p.2.2.2| * (|p.2.1.1|+2*|p.2.2.1|))).sum= |
| narrowAbsCross c v w := by |
| simp only [narrowRetainedPrefix,List.map_append,List.sum_append,narrowRetainedCommon_score, |
| narrowRetainedMiddle_score,mergedProductBranches_left,mergedProductBranches_right,narrowAbsCross,add_assoc] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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|
