| import Mathlib |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedCommon |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedExpansion |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem narrowRetainedPrefix_value (c v w : List (ℝ × (ℝ × ℝ))) |
| (hv : weightedUnderlying v=weightedUnderlying w) {f : ℝ → ℝ} (hf : BoundedDerivs f) : |
| vectorIncrementChain (narrowRetainedPrefix c v w) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) 0= |
| 3*scalarIncrementChain (weightedUnderlying c++weightedUnderlying w) f 0 := by |
| rw [narrowRetainedPrefix,vectorIncrementChain_append,narrowRetainedMiddle_value v w hf,hv] |
| have H := narrowRetainedCommon_value c (scalarIncrementChain (weightedUnderlying w) f) 0 |
| simpa only [map_zero,Prod.fst_zero,scalarIncrementChain_append] using H |
|
|
| theorem narrowRetainedPrefix_average (c v w : List (ℝ × (ℝ × ℝ))) |
| (hv : weightedUnderlying v=weightedUnderlying w) {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {H : ℝ → ℝ} (hH : Continuous H) (hHE : HasExpGrowth H) : |
| vectorIncrementAverage (narrowRetainedPrefix c 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) 0= |
| vectorIncrementAverage c (fun p : ℝ × ℝ => scalarIncrementChain (weightedUnderlying w) f p.1) |
| (fun p => |
| vectorIncrementAverage v (fun q : ℝ × ℝ => f q.1) (fun q => q.2^2) (p.1,0)* |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1) p+ |
| 2*(vectorIncrementAverage v (fun q : ℝ × ℝ => f q.1) (fun q => q.2) (p.1,0)* |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1*q.2) p)+ |
| vectorIncrementAverage w (fun q : ℝ × ℝ => f q.1) (fun q => H q.1*q.2^2) p) (0,0) := by |
| rw [narrowRetainedPrefix,vectorIncrementAverage_append,narrowRetainedMiddle_value v w hf,hv, |
| narrowRetainedMiddle_moment v w hf hH hHE] |
| exact narrowRetainedCommon_average c (scalarIncrementChain (weightedUnderlying w) f) _ 0 |
|
|
| def narrowBaseQuadratic (c v w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) : ℝ := |
| let f := scalarIncrementChain t scalarSpinTerminal |
| vectorIncrementAverage (narrowRetainedPrefix c v w) (fun p : NarrowRetainedState => f p.1.1+2*f p.2.1) |
| (fun p => rootHessian 0 f p.2.1*(p.1.2+p.2.2)^2) 0 |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
