| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.TripleSchedule |
| import OAI.Probability.SKBarriers.Locking.NarrowClockAlgebra |
|
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped BigOperators Matrix |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| abbrev NarrowMiddleIncrement := ProductIncrement (E:=ℝ × ℝ) (F:=ℝ × ℝ) |
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| def narrowCommonSchedule (δ : ℝ) (l : List (ℝ × (ℝ × ℝ))) : List (ℝ × (Fin 3 → ℝ)) := |
| l.map (fun p => (p.1/3,![p.2.1,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2])) |
|
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| def narrowMiddleVector (δ : ℝ) : NarrowMiddleIncrement → Fin 3 → ℝ |
| | .inl p => ![p.2.1,δ*p.2.2,-δ*p.2.2] |
| | .inr p => ![0,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] |
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| def narrowMiddleSchedule (δ : ℝ) (l : List NarrowMiddleIncrement) : List (ℝ × (Fin 3 → ℝ)) := |
| l.map (fun p => (productMass p,narrowMiddleVector δ p)) |
|
|
| def narrowMiddleNoise : NarrowMiddleIncrement → ℝ |
| | .inl p => p.2.2 |
| | .inr p => p.2.2 |
|
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| def narrowNoisePenalty (l : List NarrowMiddleIncrement) (K : ℝ) : ℝ := |
| chainPotentialPenalty (fun x : ℝ => x^2) (l.map (fun p => (productMass p,(narrowMiddleNoise p)^2))) K |
|
|
| theorem narrowCommonSchedule_covariance (δ : ℝ) (l : List (ℝ × (ℝ × ℝ))) (r g K : ℝ) : |
| narrowClockPerturbed r r r r r δ g g K+matrixChainCovariance (narrowCommonSchedule δ l)= |
| narrowClockPerturbed (r+rawVariance (weightedUnderlying l)) (r+rawVariance (weightedUnderlying l)) |
| (r+rawVariance (weightedUnderlying l)) (r+rawVariance (weightedUnderlying l)) |
| (r+rawVariance (weightedUnderlying l)) δ (g+weightedCross l) (g+weightedCross l) (K+weightedVariance l) := by |
| induction l generalizing r g K with |
| | nil => simp [narrowCommonSchedule,rawVariance,weightedUnderlying,weightedCross,weightedVariance] |
| | cons p l ih => |
| simp only [narrowCommonSchedule] at ih |
| simp only [narrowCommonSchedule,List.map_cons,matrixChainCovariance_cons,← add_assoc] |
| change (fun i j => narrowClockPerturbed r r r r r δ g g K i j+ |
| (![p.2.1,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] : Fin 3 → ℝ) i* |
| (![p.2.1,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] : Fin 3 → ℝ) j)+_=_ |
| rw [narrowClockPerturbed_common,ih] |
| simp only [weightedUnderlying,rawVariance,weightedCross,weightedVariance,List.map_cons,List.sum_cons,add_assoc] |
|
|
| theorem narrowCommonSchedule_penalty (δ : ℝ) (l : List (ℝ × (ℝ × ℝ))) (r g K : ℝ) : |
| matrixChainPenalty (narrowCommonSchedule δ l) (narrowClockPerturbed r r r r r δ g g K)= |
| 3*rawPenalty (weightedUnderlying l) r+4*δ^2*weightedCrossPenalty l g+ |
| 4/3*δ^4*weightedVariancePenalty l K := by |
| induction l generalizing r g K with |
| | nil => simp [narrowCommonSchedule,rawPenalty,weightedUnderlying,weightedCrossPenalty,weightedVariancePenalty,chainPotentialPenalty] |
| | cons p l ih => |
| simp only [narrowCommonSchedule] at ih |
| simp only [narrowCommonSchedule,List.map_cons,matrixChainPenalty_cons] |
| have he := narrowClockPerturbed_common r δ g K p.2.1 p.2.2 |
| change narrowClockPerturbed r r r r r δ g g K+ |
| rankOneMatrix (![p.2.1,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] : Fin 3 → ℝ)=_ at he |
| rw [he] |
| change p.1/3*(_-_)+_=_ |
| rw [narrowClockPerturbed_common_penalty,ih] |
| simp only [weightedUnderlying,rawPenalty,weightedCrossPenalty,weightedVariancePenalty,List.map_cons,chainPotentialPenalty] |
| ring |
|
|
| theorem narrowMiddleSchedule_covariance (δ : ℝ) (l : List NarrowMiddleIncrement) (a b r g y K : ℝ) : |
| narrowClockPerturbed a b b r b δ g y K+matrixChainCovariance (narrowMiddleSchedule δ l)= |
| narrowClockPerturbed (a+rawVariance (weightedUnderlying (productLeft l))) |
| (b+rawVariance (weightedUnderlying (productRight l))) (b+rawVariance (weightedUnderlying (productRight l))) |
| r (b+rawVariance (weightedUnderlying (productRight l))) δ |
| (g+weightedCross (productLeft l)) (y+weightedCross (productRight l)) |
| (K+weightedVariance (productLeft l)+weightedVariance (productRight l)) := by |
| induction l generalizing a b g y K with |
| | nil => simp [narrowMiddleSchedule,productLeft,productRight,rawVariance,weightedUnderlying,weightedCross,weightedVariance] |
| | cons p l ih => |
| simp only [narrowMiddleSchedule] at ih |
| cases p with |
| | inl p => |
| simp only [narrowMiddleSchedule,List.map_cons,matrixChainCovariance_cons,← add_assoc] |
| change (fun i j => narrowClockPerturbed a b b r b δ g y K i j+ |
| (![p.2.1,δ*p.2.2,-δ*p.2.2] : Fin 3 → ℝ) i*(![p.2.1,δ*p.2.2,-δ*p.2.2] : Fin 3 → ℝ) j)+_=_ |
| rw [narrowClockPerturbed_singleton,ih] |
| simp only [productLeft,productRight,List.filterMap_cons,weightedUnderlying,rawVariance,weightedCross,weightedVariance,List.map_cons,List.sum_cons,add_assoc] |
| | inr p => |
| simp only [narrowMiddleSchedule,List.map_cons,matrixChainCovariance_cons,← add_assoc] |
| change (fun i j => narrowClockPerturbed a b b r b δ g y K i j+ |
| (![0,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] : Fin 3 → ℝ) i*(![0,p.2.1+δ*p.2.2,p.2.1-δ*p.2.2] : Fin 3 → ℝ) j)+_=_ |
| rw [narrowClockPerturbed_pair,ih] |
| simp only [productLeft,productRight,List.filterMap_cons,weightedUnderlying,rawVariance,weightedCross,weightedVariance,List.map_cons,List.sum_cons,add_assoc] |
| congr 1 |
| ring |
|
|
| theorem narrowMiddleSchedule_penalty (δ : ℝ) (l : List NarrowMiddleIncrement) (a b r g y K : ℝ) : |
| matrixChainPenalty (narrowMiddleSchedule δ l) (narrowClockPerturbed a b b r b δ g y K)= |
| rawPenalty (weightedUnderlying (productLeft l)) a+2*rawPenalty (weightedUnderlying (productRight l)) b+ |
| 4*δ^2*weightedCrossPenalty (productLeft l) g+4*δ^2*weightedCrossPenalty (productRight l) y+ |
| 4*δ^4*narrowNoisePenalty l K := by |
| induction l generalizing a b g y K with |
| | nil => simp [narrowMiddleSchedule,productLeft,productRight,weightedUnderlying,rawPenalty, |
| weightedCrossPenalty,narrowNoisePenalty,chainPotentialPenalty] |
| | cons p l ih => |
| simp only [narrowMiddleSchedule] at ih |
| cases p with |
| | inl p => |
| simp only [narrowMiddleSchedule,List.map_cons,matrixChainPenalty_cons] |
| have he := narrowClockPerturbed_singleton a b r δ g y K p.2.1 p.2.2 |
| change narrowClockPerturbed a b b r b δ g y K+rankOneMatrix (narrowMiddleVector δ (.inl p))=_ at he |
| rw [he] |
| change p.1*(_-_)+_=_ |
| rw [narrowClockPerturbed_singleton_penalty,ih] |
| simp only [productLeft,productRight,List.filterMap_cons,weightedUnderlying,rawPenalty, |
| weightedCrossPenalty,narrowNoisePenalty,narrowMiddleNoise,productMass,List.map_cons,chainPotentialPenalty] |
| ring |
| | inr p => |
| simp only [narrowMiddleSchedule,List.map_cons,matrixChainPenalty_cons] |
| have he := narrowClockPerturbed_pair a b r δ g y K p.2.1 p.2.2 |
| change narrowClockPerturbed a b b r b δ g y K+rankOneMatrix (narrowMiddleVector δ (.inr p))=_ at he |
| rw [he] |
| change p.1/2*(_-_)+_=_ |
| rw [narrowClockPerturbed_pair_penalty,ih] |
| simp only [productLeft,productRight,List.filterMap_cons,weightedUnderlying,rawPenalty, |
| weightedCrossPenalty,narrowNoisePenalty,narrowMiddleNoise,productMass,List.map_cons,chainPotentialPenalty] |
| ring |
|
|
| end SK.Analytic |
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| end |
| end |
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| end OAI |
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