| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.ConstantWeightStats |
| import OAI.Probability.SKBarriers.Scalar.ScalarTentGeometry |
| import OAI.Probability.SKBarriers.Hierarchy.ListPrefixTake |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| namespace SK.Analytic |
|
|
| abbrev fourBlockTent (κ : ℝ) (b l j k : List (ℝ × ℝ)) : List (ℝ × (ℝ × ℝ)) := |
| (zeroWeightChain b++constantWeightChain κ l)++(constantWeightChain (-κ) j++zeroWeightChain k) |
|
|
| @[simp] theorem fourBlockTent_length (κ : ℝ) (b l j k : List (ℝ × ℝ)) : |
| (fourBlockTent κ b l j k).length=b.length+l.length+j.length+k.length := by |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,List.length_append,List.length_map] |
| omega |
|
|
| abbrev fourBlockLeft (κ : ℝ) (b l j k : List (ℝ × ℝ)) : Fin ((fourBlockTent κ b l j k).length+1) := |
| ⟨b.length,by rw [fourBlockTent_length]; omega⟩ |
| abbrev fourBlockMiddle (κ : ℝ) (b l j k : List (ℝ × ℝ)) : Fin ((fourBlockTent κ b l j k).length+1) := |
| ⟨b.length+l.length,by rw [fourBlockTent_length]; omega⟩ |
| abbrev fourBlockRight (κ : ℝ) (b l j k : List (ℝ × ℝ)) : Fin ((fourBlockTent κ b l j k).length+1) := |
| ⟨b.length+l.length+j.length,by rw [fourBlockTent_length]; omega⟩ |
|
|
| theorem fourBlockTent_vector {κ d : ℝ} (hd : d≠0) (hκ : κ*d=1) (b l j k : List (ℝ × ℝ)) : |
| (fun i => ((fourBlockTent κ b l j k).get i).2.2)= |
| scalarTentVector (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).2.1) |
| (fourBlockLeft κ b l j k) (fourBlockMiddle κ b l j k) (fourBlockRight κ b l j k) d := by |
| funext i |
| simp only [List.get_eq_getElem] |
| have hfactor : κ=1/d := (eq_div_iff hd).mpr hκ |
| unfold scalarTentVector scalarPrefixVector |
| simp only [] |
| by_cases hb : i.val<b.length |
| · have hbl : i.val<b.length+l.length := by omega |
| have hblj : i.val<b.length+l.length+j.length := by omega |
| simp only [hb,hbl,hblj,ite_true] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,List.getElem_append, |
| List.length_append,List.length_map,hbl,hb,dite_true,List.getElem_map] |
| ring |
| · by_cases hl : i.val<b.length+l.length |
| · have hblj : i.val<b.length+l.length+j.length := by omega |
| have hil : i.val-b.length<l.length := by omega |
| simp only [hb,hl,hblj,ite_true,ite_false] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,List.getElem_append, |
| List.length_append,List.length_map,hl,hb,dite_true,dite_false,List.getElem_map] |
| apply (eq_div_iff hd).mpr |
| linear_combination l[i.val-b.length].2*hκ |
| · by_cases hj : i.val<b.length+l.length+j.length |
| · have hij : i.val-(b.length+l.length)<j.length := by omega |
| simp only [hb,hl,hj,ite_true,ite_false] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,List.getElem_append, |
| List.length_append,List.length_map,hl,hb,hij,dite_true,dite_false,List.getElem_map] |
| apply (eq_div_iff hd).mpr |
| linear_combination -j[i.val-(b.length+l.length)].2*hκ |
| · have hij : ¬i.val-(b.length+l.length)<j.length := by omega |
| simp only [hb,hl,hj,ite_false] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,List.getElem_append, |
| List.length_append,List.length_map,hl,hb,hij,dite_false,List.getElem_map] |
| ring |
|
|
| theorem fourBlockTent_left_variance (κ : ℝ) (b l j k : List (ℝ × ℝ)) : |
| scalarPrefixVariance (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).2.1) |
| (fourBlockLeft κ b l j k)=rawVariance b := by |
| rw [scalarPrefixVariance_list (fourBlockTent κ b l j k) (fun p => p.2.1)] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,rawVariance, |
| List.take_append,← List.map_take,List.length_map,List.map_append,List.sum_append,List.map_map,Function.comp_def] |
| simp (disch := (simp only [List.length_map]; omega)) [Nat.sub_eq_zero_of_le (show b.length≤b.length+l.length by omega)] |
|
|
| theorem fourBlockTent_middle_variance (κ : ℝ) (b l j k : List (ℝ × ℝ)) : |
| scalarPrefixVariance (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).2.1) |
| (fourBlockMiddle κ b l j k)=rawVariance b+rawVariance l := by |
| rw [scalarPrefixVariance_list (fourBlockTent κ b l j k) (fun p => p.2.1)] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,rawVariance, |
| List.take_append,← List.map_take,List.length_map,List.map_append,List.sum_append,List.map_map,Function.comp_def] |
| simp (disch := (simp only [List.length_map]; omega)) [List.take_of_length_le] |
|
|
| theorem fourBlockTent_right_variance (κ : ℝ) (b l j k : List (ℝ × ℝ)) : |
| scalarPrefixVariance (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).2.1) |
| (fourBlockRight κ b l j k)=rawVariance b+rawVariance l+rawVariance j := by |
| rw [scalarPrefixVariance_list (fourBlockTent κ b l j k) (fun p => p.2.1)] |
| simp only [fourBlockTent,zeroWeightChain,constantWeightChain,rawVariance, |
| List.take_append,← List.map_take,List.length_map,List.map_append,List.sum_append,List.map_map,Function.comp_def] |
| simp (disch := (simp only [List.length_map]; omega)) [List.take_of_length_le] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
