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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 | 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
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