| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.WeightedListBridge |
| import OAI.Probability.SKBarriers.Hierarchy.RootCovariance |
| import OAI.Probability.SKBarriers.Scalar.ScalarListBounds |
| import OAI.Probability.SKBarriers.Replicas.TripleFiniteExpansion |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| def tripleSusceptibility (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) : ℝ := |
| scalarIncrementAverage (c++weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal) |
| (rootHessian 0 (scalarIncrementChain t scalarSpinTerminal)) 0 |
|
|
| def tripleCommonError (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) : ℝ := |
| 2*weightedMomentBound w*((susceptibilityLipschitzConstant:ℝ)+2*Real.exp 2)* |
| Real.sqrt (2*rawVariance c+4*(rawVariance c)^2) |
|
|
| theorem tripleBaseQuadratic_decoupling (c : List (ℝ × ℝ)) (w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) |
| (hm : ∀ p∈c++weightedUnderlying w++t,p.1∈Icc (0:ℝ) 1) |
| (hs : (c++weightedUnderlying w++t).Pairwise (fun p q => p.1≤q.1)) : |
| |tripleBaseQuadratic c w t-tripleSusceptibility c w t* |
| vectorIncrementAverage (zeroWeightChain c++w++zeroWeightChain t) |
| (fun p : ℝ × ℝ => scalarSpinTerminal p.1) (fun p => p.2^2) (0,0)|≤tripleCommonError c w := by |
| have hmc : ∀ p∈c,p.1∈Icc (0:ℝ) 1 := fun p hp => hm p (List.mem_append_left _ (List.mem_append_left _ hp)) |
| have hmw : ∀ p∈weightedUnderlying w,p.1∈Icc (0:ℝ) 1 := fun p hp => hm p (List.mem_append_left _ (List.mem_append_right _ hp)) |
| have hmt : ∀ p∈t,p.1∈Icc (0:ℝ) 1 := fun p hp => hm p (List.mem_append_right _ hp) |
| have hs0 := List.pairwise_append.mp hs |
| have hs1 := List.pairwise_append.mp hs0.1 |
| have hww : ∀ p∈w,p.1∈Icc (0:ℝ) 1 := fun p hp => hmw _ (List.mem_map.mpr ⟨p,hp,rfl⟩) |
| have hsw : w.Pairwise (fun p q => p.1≤q.1) := by |
| simpa only [weightedUnderlying,List.pairwise_map] using hs1.2.1 |
| let f := scalarIncrementChain t scalarSpinTerminal |
| let F := scalarIncrementChain (weightedUnderlying w) f |
| let A := scalarIncrementAverage (weightedUnderlying w) f (rootHessian 0 f) |
| let B := fun x => vectorIncrementAverage w (fun p : ℝ × ℝ => f p.1) (fun p => p.2^2) (x,0) |
| have hf : BoundedDerivs f := scalarIncrementChain_regular _ scalarSpinTerminal_regular |
| have hfL : LipschitzWith 1 f := scalarIncrementChain_lipschitz t (fun p hp => (hmt p hp).1) |
| scalarSpinTerminal_regular scalarSpinTerminal_lipschitz |
| have hF : BoundedDerivs F := scalarIncrementChain_regular _ hf |
| have hFL : LipschitzWith 1 F := scalarIncrementChain_lipschitz _ (fun p hp => (hmw p hp).1) hf hfL |
| have hfc : ScalarSpinConvex f := scalarIncrementChain_spin_convex t hmt |
| have hA : BoundedScalar A := scalarIncrementAverage_bounded _ hf (hfc.hessian_bounded hf) |
| have hB : BoundedScalar B := weightedAverage_bounded w hww hsw hf hfL |
| have hAL (x : ℝ) : |A x-A 0|≤((susceptibilityLipschitzConstant:ℝ)+2*Real.exp 2)*|x| := by |
| have H := scalarHierarchyAverage_spatial_bound (weightedUnderlying w).length |
| (fun i => ((weightedUnderlying w).get i).1) (fun i => ((weightedUnderlying w).get i).2) |
| (fun i => hmw _ (List.get_mem _ i)) (mass_get_monotone hs1.2.1) hf hfL |
| (scalarIncrementChain_hessian_lipschitz t hmt hs0.2.1) (C:=1) (fun z => (hfc.bounds z).2) x 0 |
| simpa only [← scalarIncrementAverage_get,NNReal.coe_one,mul_one,sub_zero,A] using H |
| have H := scalarHierarchyAverage_covariance_small c.length (fun i => (c.get i).1) (fun i => (c.get i).2) |
| (fun i => hmc _ (List.get_mem c i)) (mass_get_monotone hs1.1) hF hFL hA hB |
| (weightedMomentBound_nonneg w) (show 0≤(susceptibilityLipschitzConstant:ℝ)+2*Real.exp 2 by positivity) |
| (weightedAverage_abs_le w hww hsw hf hfL) hAL |
| rw [sum_get_map c (fun p => p.2^2)] at H |
| simp only [← scalarIncrementAverage_get] at H |
| rw [weightedFull_average c w t scalarSpinTerminal_regular] |
| unfold tripleBaseQuadratic tripleSusceptibility tripleCommonError |
| rw [scalarIncrementAverage_append] |
| simpa only [A,B,F,f,rawVariance,mul_comm] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
