File size: 5,523 Bytes
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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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
|