| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.ProductMerge |
| import OAI.Probability.SKBarriers.Replicas.TripleRetained |
| import OAI.Probability.SKBarriers.Hierarchy.IncrementListAnalytic |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem list_commonBranch_value (c : List (ℝ × ℝ)) (f : ℝ → ℝ) (x : ℝ) : |
| vectorIncrementChain (tripleRetainedCommon c) |
| (fun p : TripleRetainedState => f p.1.1+2*f p.2) ((x,0),x)=3*scalarIncrementChain c f x := by |
| have H := commonBranch_value c.length (fun i => (c.get i).1) (fun i => (c.get i).2) f x |
| rw [← vectorIncrementChain_ofFn,← scalarIncrementChain_ofFn] at H |
| have h1 : List.ofFn (fun i : Fin c.length => ((c.get i).1/3,(((c.get i).2,0),(c.get i).2)))= |
| tripleRetainedCommon c := by |
| unfold tripleRetainedCommon |
| simpa only [List.ofFn_get,Function.comp_def] using |
| (List.map_ofFn (f:=c.get) (g:=fun p : ℝ × ℝ => (p.1/3,((p.2,0),p.2)))).symm |
| simpa only [h1,Prod.mk.eta,List.ofFn_get] using H |
|
|
| theorem list_commonBranch_average (c : List (ℝ × ℝ)) (f b : ℝ → ℝ) (a : ℝ × ℝ → ℝ) (x : ℝ) : |
| vectorIncrementAverage (tripleRetainedCommon c) |
| (fun p : TripleRetainedState => f p.1.1+2*f p.2) (fun p => a p.1*b p.2) ((x,0),x)= |
| scalarIncrementAverage c f (fun y => a (y,0)*b y) x := by |
| have H := commonBranch_average c.length (fun i => (c.get i).1) (fun i => (c.get i).2) f b a x |
| rw [← vectorIncrementAverage_ofFn,← scalarIncrementAverage_ofFn] at H |
| have h1 : List.ofFn (fun i : Fin c.length => ((c.get i).1/3,(((c.get i).2,0),(c.get i).2)))= |
| tripleRetainedCommon c := by |
| unfold tripleRetainedCommon |
| simpa only [List.ofFn_get,Function.comp_def] using |
| (List.map_ofFn (f:=c.get) (g:=fun p : ℝ × ℝ => (p.1/3,((p.2,0),p.2)))).symm |
| simpa only [h1,Prod.mk.eta,List.ofFn_get] using H |
|
|
| theorem weightedBranch_value (w : List (ℝ × (ℝ × ℝ))) (f : ℝ → ℝ) : |
| vectorIncrementChain w (fun p : ℝ × ℝ => f p.1)=fun p => scalarIncrementChain (weightedUnderlying w) f p.1 := by |
| have H := vectorIncrementChain_pullback (ContinuousLinearMap.fst ℝ ℝ ℝ) w f |
| rw [vectorIncrementChain_real] at H |
| exact H |
|
|
| theorem tripleRetainedMiddle_value (w : List (ℝ × (ℝ × ℝ))) {f : ℝ → ℝ} (hf : BoundedDerivs f) : |
| vectorIncrementChain (tripleRetainedMiddle (mergedProductBranches w (weightedUnderlying w))) |
| (fun p : TripleRetainedState => f p.1.1+2*f p.2)= |
| fun p => scalarIncrementChain (weightedUnderlying w) f p.1.1+2*scalarIncrementChain (weightedUnderlying w) f p.2 := by |
| have hF : BoundedDerivs (fun q : ℝ × ℝ => f q.1) := hf.compCLM (ContinuousLinearMap.fst ℝ ℝ ℝ) |
| change productBranchChain _ (fun p => (fun q : ℝ × ℝ => f q.1) p.1+2*f p.2)=_ |
| rw [productBranchChain_factor _ hF hf, |
| mergedProductBranches_left,mergedProductBranches_right,weightedBranch_value,vectorIncrementChain_real] |
|
|
| theorem tripleRetainedMiddle_average (w : List (ℝ × (ℝ × ℝ))) {f : ℝ → ℝ} (hf : BoundedDerivs f) (g : ℝ → ℝ) : |
| vectorIncrementAverage (tripleRetainedMiddle (mergedProductBranches w (weightedUnderlying w))) |
| (fun p : TripleRetainedState => f p.1.1+2*f p.2) (fun p => g p.2*p.1.2^2)= |
| fun p => vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => q.2^2) p.1* |
| scalarIncrementAverage (weightedUnderlying w) f g p.2 := by |
| have he : (fun p : TripleRetainedState => g p.2*p.1.2^2)=fun p => p.1.2^2*g p.2 := by funext p; ring |
| rw [he] |
| have hF : BoundedDerivs (fun q : ℝ × ℝ => f q.1) := hf.compCLM (ContinuousLinearMap.fst ℝ ℝ ℝ) |
| change productBranchAverage _ (fun p => (fun q : ℝ × ℝ => f q.1) p.1+2*f p.2) (fun p => (fun q : ℝ × ℝ => q.2^2) p.1*g p.2)=_ |
| rw [productBranchAverage_factor _ hF hf (fun q : ℝ × ℝ => q.2^2) g, |
| mergedProductBranches_left,mergedProductBranches_right,vectorIncrementAverage_real] |
|
|
| def tripleRetainedPrefix (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) : List (ℝ × TripleRetainedState) := |
| tripleRetainedCommon c++tripleRetainedMiddle (mergedProductBranches w (weightedUnderlying w)) |
|
|
| theorem tripleRetainedPrefix_value (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) : |
| vectorIncrementChain (tripleRetainedPrefix c w) (fun p => f p.1.1+2*f p.2) 0= |
| 3*scalarIncrementChain (c++weightedUnderlying w) f 0 := by |
| rw [tripleRetainedPrefix,vectorIncrementChain_append,tripleRetainedMiddle_value w hf,scalarIncrementChain_append] |
| exact list_commonBranch_value c _ 0 |
|
|
| theorem tripleRetainedPrefix_average (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) (g : ℝ → ℝ) : |
| vectorIncrementAverage (tripleRetainedPrefix c w) (fun p => f p.1.1+2*f p.2) (fun p => g p.2*p.1.2^2) 0= |
| scalarIncrementAverage c (scalarIncrementChain (weightedUnderlying w) f) |
| (fun x => vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => q.2^2) (x,0)* |
| scalarIncrementAverage (weightedUnderlying w) f g x) 0 := by |
| rw [tripleRetainedPrefix,vectorIncrementAverage_append,tripleRetainedMiddle_value w hf, |
| tripleRetainedMiddle_average w hf g] |
| exact list_commonBranch_average c _ _ _ 0 |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
