| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.TimeChainOnAnalytic |
| import OAI.Probability.SKBarriers.Scalar.SuffixPartitionSusceptibility |
| import OAI.Probability.SKBarriers.Locking.NarrowTimeBlocks |
| import OAI.Probability.SKBarriers.Locking.NarrowMomentBound |
| import OAI.Probability.SKBarriers.Parisi.CDFJointBridge |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem narrowJointMoment_eq (c w : List (ℝ × (ℝ × ℝ))) (t : List (ℝ × ℝ)) : |
| narrowJointMoment c w t=scalarIncrementAverage (weightedUnderlying c) |
| (scalarIncrementChain (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal)) |
| (fun x => rootHessian 0 (scalarIncrementChain (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal)) x* |
| scalarIncrementAverage (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal) |
| (rootHessian 0 (scalarIncrementChain t scalarSpinTerminal)) x) 0 := by |
| dsimp only [narrowJointMoment] |
| simp_rw [weightedBranch_average] |
| simp_rw [mul_comm] |
| exact congrFun (weightedBranch_average c |
| (scalarIncrementChain (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal)) |
| (fun x => scalarIncrementAverage (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal) |
| (rootHessian 0 (scalarIncrementChain t scalarSpinTerminal)) x* |
| rootHessian 0 (scalarIncrementChain (weightedUnderlying w) (scalarIncrementChain t scalarSpinTerminal)) x)) (0,0) |
|
|
| namespace NarrowTimeBlocks |
| variable {α : ℝ → ℝ} {r h q : ℝ} (B : NarrowTimeBlocks α r h q) |
|
|
| theorem full_mass (a : ℝ) : ∀ p∈weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t,p.1∈Icc (0:ℝ) 1 := by |
| rw [← B.full_raw a] |
|
|
| theorem full_sorted (hm : Monotone α) (a : ℝ) : |
| (weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t).Pairwise (fun p q => p.1≤q.1) := by |
| rw [← B.full_raw a] |
|
|
| theorem scalar_parisi (β : ℝ) (ha : ∀ z,α z∈Icc (0:ℝ) 1) (hm : Monotone α) (a : ℝ) : |
| scalarIncrementChain (scaleIncrementChain β (weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t)) scalarSpinTerminal 0- |
| β^2/4*rawPenalty (weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t) 0=scalarCDFParisi β α := by |
| rw [← B.full_raw a,B.full.scalar_value β ha hm (by norm_num),B.full.penalty hm] |
| simp only [sub_zero,Real.toNNReal_one,scalarCDFParisi,intervalIntegral.integral_of_le zero_le_one, |
| ← integral_Icc_eq_integral_Ioc] |
| ring |
|
|
| theorem scalar_susceptibility (β : ℝ) (ha : ∀ z,α z∈Icc (0:ℝ) 1) (hm : Monotone α) |
| (hq : q∈Icc (0:ℝ) 1) (a : ℝ) : |
| narrowSusceptibility (scaleIncrementChain β (B.c a)) (scaleIncrementChain β (B.w a)) (scaleIncrementChain β B.t)= |
| scalarCDFSusceptibilityAverage β α q := by |
| rw [narrowSusceptibility_eq,weightedUnderlying_append,weightedUnderlying_scale,weightedUnderlying_scale, |
| ← scaleIncrementChain_append,← B.front_raw a] |
| exact B.front.scalar_susceptibility β ha hm hq B.tail |
| |
| theorem scalar_joint (β : ℝ) (ha : ∀ z,α z∈Icc (0:ℝ) 1) (hm : Monotone α) |
| (hr : 0≤r) (hrq : r≤q) (hq : q≤1) (a : ℝ) : |
| narrowJointMoment (scaleIncrementChain β (B.c a)) (scaleIncrementChain β (B.w a)) (scaleIncrementChain β B.t)= |
| scalarCDFJointSusceptibility β α r q := by |
| rw [narrowJointMoment_eq,weightedUnderlying_scale,weightedUnderlying_scale,← B.common_raw a,← B.middle_raw a] |
| simp only [t,scalarIncrementChain_rawTimeChain,scalarIncrementAverage_rawTimeChain] |
| exact timeChainJointHessian_eq_cdf β ha hm hr hrq hq B.common B.middle B.tail |
| |
| theorem tail_area (hm : Monotone α) : rawArea B.t=∫ x in q..1,α x := B.tail.area hm |
|
|
| theorem right_area (hm : Monotone α) : rawArea B.j=∫ x in r..r+h,α x := B.right.area hm |
|
|
| end NarrowTimeBlocks |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
