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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 | 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
end OAI
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