| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleBaseFactor |
| import OAI.Probability.SKBarriers.Scalar.VectorAverageAlgebra |
| import OAI.Probability.SKBarriers.Locking.NarrowSchedule |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| abbrev NarrowRetainedState := (ℝ × ℝ) × (ℝ × ℝ) |
|
|
| def narrowRetainedCommon (c : List (ℝ × (ℝ × ℝ))) : List (ℝ × NarrowRetainedState) := |
| c.map (fun p => (p.1/3,((p.2.1,0),(p.2.1,p.2.2)))) |
|
|
| def narrowRetainedMiddle (l : List NarrowMiddleIncrement) : List (ℝ × NarrowRetainedState) := |
| l.map (fun p => (productMass p,productVector p)) |
|
|
| theorem narrowRetainedMiddle_value (v w : List (ℝ × (ℝ × ℝ))) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) : |
| vectorIncrementChain (narrowRetainedMiddle (mergedProductBranches v w)) |
| (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1)= |
| fun p => scalarIncrementChain (weightedUnderlying v) f p.1.1+ |
| 2*scalarIncrementChain (weightedUnderlying w) f p.2.1 := by |
| have hF : BoundedDerivs (fun p : ℝ × ℝ => f p.1) := hf.compCLM (ContinuousLinearMap.fst ℝ ℝ ℝ) |
| change productBranchChain _ (fun p => (fun q : ℝ × ℝ => f q.1) p.1+2*(fun q : ℝ × ℝ => f q.1) p.2)=_ |
| rw [productBranchChain_factor _ hF hF,mergedProductBranches_left,mergedProductBranches_right, |
| weightedBranch_value,weightedBranch_value] |
|
|
| theorem narrowRetainedMiddle_moment (v w : List (ℝ × (ℝ × ℝ))) |
| {f H : ℝ → ℝ} (hf : BoundedDerivs f) (hH : Continuous H) (hHE : HasExpGrowth H) : |
| vectorIncrementAverage (narrowRetainedMiddle (mergedProductBranches v w)) |
| (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) |
| (fun p => H p.2.1*(p.1.2+p.2.2)^2)= |
| fun p => |
| vectorIncrementAverage v (fun q : ℝ × ℝ => f q.1) (fun q => q.2^2) p.1* |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1) p.2+ |
| 2*(vectorIncrementAverage v (fun q : ℝ × ℝ => f q.1) (fun q => q.2) p.1* |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1*q.2) p.2)+ |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1*q.2^2) p.2 := by |
| let A : NarrowRetainedState →L[ℝ] ℝ := (ContinuousLinearMap.snd ℝ ℝ ℝ).comp (ContinuousLinearMap.fst ℝ (ℝ × ℝ) (ℝ × ℝ)) |
| let B : NarrowRetainedState →L[ℝ] ℝ := (ContinuousLinearMap.snd ℝ ℝ ℝ).comp (ContinuousLinearMap.snd ℝ (ℝ × ℝ) (ℝ × ℝ)) |
| let X : NarrowRetainedState →L[ℝ] ℝ := (ContinuousLinearMap.fst ℝ ℝ ℝ).comp (ContinuousLinearMap.snd ℝ (ℝ × ℝ) (ℝ × ℝ)) |
| let V : NarrowRetainedState →L[ℝ] ℝ := (ContinuousLinearMap.fst ℝ ℝ ℝ).comp (ContinuousLinearMap.fst ℝ (ℝ × ℝ) (ℝ × ℝ)) |
| let L := narrowRetainedMiddle (mergedProductBranches v w) |
| let F := fun p : NarrowRetainedState => f p.1.1+2*f p.2.1 |
| have hF : BoundedDerivs F := (hf.compCLM V).add ((hf.compCLM X).const_mul 2) |
| have hc1 : Continuous (fun p : NarrowRetainedState => p.1.2^2*H p.2.1) := (A.continuous.pow 2).mul (hH.comp X.continuous) |
| have he1 : HasExpGrowth (fun p : NarrowRetainedState => p.1.2^2*H p.2.1) := ((HasExpGrowth.linear A).pow 2).mul (hHE.compCLM X) |
| have hc2 : Continuous (fun p : NarrowRetainedState => p.1.2*(H p.2.1*p.2.2)) := A.continuous.mul ((hH.comp X.continuous).mul B.continuous) |
| have he2 : HasExpGrowth (fun p : NarrowRetainedState => p.1.2*(H p.2.1*p.2.2)) := (HasExpGrowth.linear A).mul ((hHE.compCLM X).mul (HasExpGrowth.linear B)) |
| have hc3 : Continuous (fun p : NarrowRetainedState => H p.2.1*p.2.2^2) := (hH.comp X.continuous).mul (B.continuous.pow 2) |
| have he3 : HasExpGrowth (fun p : NarrowRetainedState => H p.2.1*p.2.2^2) := (hHE.compCLM X).mul ((HasExpGrowth.linear B).pow 2) |
| have he : (fun p : NarrowRetainedState => H p.2.1*(p.1.2+p.2.2)^2)= |
| fun p => (p.1.2^2*H p.2.1+2*(p.1.2*(H p.2.1*p.2.2)))+H p.2.1*p.2.2^2 := by funext p; ring |
| have hc2m : Continuous (fun p : NarrowRetainedState => 2*(p.1.2*(H p.2.1*p.2.2))) := hc2.const_mul 2 |
| have he2m : HasExpGrowth (fun p : NarrowRetainedState => 2*(p.1.2*(H p.2.1*p.2.2))) := (HasExpGrowth.const 2).mul he2 |
| have hc12 : Continuous (fun p : NarrowRetainedState => p.1.2^2*H p.2.1+2*(p.1.2*(H p.2.1*p.2.2))) := hc1.add hc2m |
| have he12 : HasExpGrowth (fun p : NarrowRetainedState => p.1.2^2*H p.2.1+2*(p.1.2*(H p.2.1*p.2.2))) := he1.add he2m |
| rw [he,vectorIncrementAverage_add L hF hc12 hc3 he12 he3, |
| vectorIncrementAverage_add L hF hc1 hc2m he1 he2m, |
| vectorIncrementAverage_const_mul] |
| have hsf : BoundedDerivs (fun p : ℝ × ℝ => f p.1) := hf.compCLM (ContinuousLinearMap.fst ℝ ℝ ℝ) |
| have hfac (a b : ℝ × ℝ → ℝ) := productBranchAverage_factor (mergedProductBranches v w) hsf hsf a b |
| simp only [mergedProductBranches_left,mergedProductBranches_right] at hfac |
| have hfac3 := hfac (fun _ => 1) (fun q => H q.1*q.2^2) |
| simp only [one_mul,vectorIncrementAverage_const v hsf 1] at hfac3 |
| change (fun p => (productBranchAverage (mergedProductBranches v w) F (fun p : NarrowRetainedState => p.1.2^2*H p.2.1) p+ |
| 2*productBranchAverage (mergedProductBranches v w) F (fun p : NarrowRetainedState => p.1.2*(H p.2.1*p.2.2)) p)+ |
| productBranchAverage (mergedProductBranches v w) F (fun p : NarrowRetainedState => H p.2.1*p.2.2^2) p)=_ |
| dsimp only [F] |
| rw [hfac (fun q => q.2^2) (fun q => H q.1),hfac (fun q => q.2) (fun q => H q.1*q.2),hfac3] |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
