| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleBaseFactor |
| import OAI.Probability.SKBarriers.Replicas.TripleScheduleMass |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem sum_get_map {α : Type} (l : List α) (f : α → ℝ) : |
| (∑ i : Fin l.length,f (l.get i))=(l.map f).sum := by |
| rw [← List.sum_ofFn] |
| simpa only [List.ofFn_get,Function.comp_def] using |
| congrArg List.sum (List.map_ofFn (f:=l.get) (g:=f)).symm |
|
|
| theorem mass_get_monotone {E : Type} {l : List (ℝ × E)} |
| (hs : l.Pairwise (fun p q => p.1≤q.1)) : Monotone (fun i : Fin l.length => (l.get i).1) := by |
| have H := hs |
| rw [← List.ofFn_get l,List.pairwise_ofFn] at H |
| intro i j hij |
| rcases lt_or_eq_of_le hij with hij|rfl |
| · exact H hij |
| · exact le_rfl |
|
|
| theorem tripleRetainedPrefix_image (δ : ℝ) (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) : |
| (tripleRetainedPrefix c w).map (fun p => (p.1,tripleRetainedEmbedding δ p.2))=tripleSchedule δ c w [] := by |
| simp only [tripleRetainedPrefix,List.map_append,tripleRetainedCommon_image,tripleRetainedMiddle_image, |
| tripleSchedule,tripleTailSchedule,List.flatMap_nil,List.append_nil] |
|
|
| theorem tripleRetainedPrefix_mass (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) |
| (hm : ∀ p∈c++weightedUnderlying w,p.1∈Icc (0:ℝ) 1) : |
| ∀ p∈tripleRetainedPrefix c w,p.1∈Icc (0:ℝ) 1 := by |
| intro p hp |
| have H := tripleSchedule_mass_bounds 0 c w [] (by simpa only [List.append_nil] using hm) |
| rw [← tripleRetainedPrefix_image] at H |
| exact H (p.1,tripleRetainedEmbedding 0 p.2) (List.mem_map.mpr ⟨p,hp,rfl⟩) |
|
|
| theorem tripleRetainedPrefix_monotone (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) |
| (hm : ∀ p∈c++weightedUnderlying w,0≤p.1) |
| (hs : (c++weightedUnderlying w).Pairwise (fun p q => p.1≤q.1)) : |
| (tripleRetainedPrefix c w).Pairwise (fun p q => p.1≤q.1) := by |
| have H := tripleSchedule_monotone 0 c w [] (by simpa only [List.append_nil] using hm) |
| (by simpa only [List.append_nil] using hs) |
| rw [← tripleRetainedPrefix_image,List.pairwise_map] at H |
| exact H |
|
|
| def weightedAbsCross (w : List (ℝ × (ℝ × ℝ))) : ℝ := (w.map (fun p => |p.2.2| * |p.2.1|)).sum |
|
|
| theorem tripleRetainedMiddle_variance (l : List TripleMiddleIncrement) : |
| ((tripleRetainedMiddle l).map (fun p => p.2.1.2^2)).sum=weightedVariance (productLeft l) := by |
| induction l with |
| | nil => rfl |
| | cons p l ih => |
| cases p <;> (simp only [tripleRetainedMiddle,List.map_cons,List.map_map,Function.comp_def, |
| List.sum_cons,productVector,productLeft,List.filterMap_cons,weightedVariance,List.map_cons, |
| List.sum_cons] at ih ⊢; rw [ih] <;> simp) |
|
|
| theorem tripleRetainedMiddle_score (l : List TripleMiddleIncrement) : |
| ((tripleRetainedMiddle l).map (fun p => |p.2.1.2| * (|p.2.1.1| + 2*|p.2.2|))).sum= |
| weightedAbsCross (productLeft l) := by |
| induction l with |
| | nil => rfl |
| | cons p l ih => |
| cases p <;> (simp only [tripleRetainedMiddle,List.map_cons,List.map_map,Function.comp_def, |
| List.sum_cons,productVector,productLeft,List.filterMap_cons,weightedAbsCross,List.map_cons,List.sum_cons, |
| abs_zero,mul_zero,add_zero] at ih ⊢; rw [ih] <;> simp) |
|
|
| theorem tripleRetainedPrefix_variance (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) : |
| ((tripleRetainedPrefix c w).map (fun p => p.2.1.2^2)).sum=weightedVariance w := by |
| simp only [tripleRetainedPrefix,List.map_append,List.sum_append,tripleRetainedMiddle_variance, |
| mergedProductBranches_left,tripleRetainedCommon,List.map_map,Function.comp_def,zero_pow (by norm_num : 2≠0), |
| ] |
| simp |
|
|
| theorem tripleRetainedPrefix_score (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) : |
| ((tripleRetainedPrefix c w).map (fun p => |p.2.1.2| * (|p.2.1.1| + 2*|p.2.2|))).sum=weightedAbsCross w := by |
| simp only [tripleRetainedPrefix,List.map_append,List.sum_append,tripleRetainedMiddle_score, |
| mergedProductBranches_left,tripleRetainedCommon,List.map_map,Function.comp_def,abs_zero,zero_mul, |
| ] |
| simp |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
