| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.SuffixPartitionSusceptibility |
| import OAI.Probability.SKBarriers.Hierarchy.ConstantWeightMean |
| import OAI.Probability.SKBarriers.Parisi.CDFJointSusceptibility |
| import OAI.Probability.SKBarriers.Locking.NarrowScheduleScale |
| import OAI.Probability.SKBarriers.Locking.NarrowBlockStats |
| import OAI.Probability.SKBarriers.Locking.NarrowTimeAnalytic |
| import OAI.Probability.SKBarriers.Locking.FourBlockAtom |
| import OAI.Probability.SKBarriers.Locking.FourBlockQuadratic |
| import OAI.Probability.SKBarriers.Locking.FourBlockMass |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| def narrowCDFMean (β : ℝ) (α : ℝ → ℝ) (r h θ : ℝ) : ℝ := β^2*(θ/h)*(∫ x in r..r+h,α x) |
|
|
| namespace NarrowTimeBlocks |
| variable {α : ℝ → ℝ} {r h q : ℝ} (C : NarrowTimeBlocks α r h q) |
|
|
| theorem scaled_v_mean (β : ℝ) (hα : Monotone α) (θ : ℝ) : |
| weightedMean (scaleIncrementChain β (C.v θ))=narrowCDFMean β α r h θ := by |
| rw [weightedMean_scale,C.v_mean,C.right_area hα] |
| unfold narrowCDFMean; ring |
|
|
| theorem quadratic_le (β : ℝ) (hα : ∀ z,α z∈Icc (0:ℝ) 1) (hmα : Monotone α) |
| (hr : 0≤r) (hrq : r≤q) (hq : q≤1) (hh : 0<h) {a θ : ℝ} (ha : 0<a) (hθ : 0≤θ) : |
| narrowBaseQuadratic (scaleIncrementChain β (C.c a)) (scaleIncrementChain β (C.v θ)) |
| (scaleIncrementChain β (C.w a)) (scaleIncrementChain β C.t)≤ |
| β^2*((2*a^2+θ^2)/h)*scalarCDFSusceptibilityAverage β α q+ |
| 2*narrowCDFMean β α r h θ*(β^2*a)*scalarCDFJointSusceptibility β α r q+ |
| narrowTentError (narrowCDFMean β α r h θ) (β^2*a) (β^2*θ) (α (r+h)-α (r-h)) := by |
| let b := scaleIncrementChain β C.b |
| let l := scaleIncrementChain β C.l |
| let j := scaleIncrementChain β C.j |
| let k := scaleIncrementChain β C.k |
| let t := scaleIncrementChain β C.t |
| let κ := a/h |
| let η := θ/h |
| have hc : scaleIncrementChain β (C.c a)=fourBlockCommon κ b l := by |
| simp only [c,scaleIncrementChain_append,zeroWeightChain_scale,constantWeightChain_scale]; rfl |
| have hv : scaleIncrementChain β (C.v θ)=fourBlockFuture η j k := by |
| simp only [v,scaleIncrementChain_append,zeroWeightChain_scale,constantWeightChain_scale]; rfl |
| have hw : scaleIncrementChain β (C.w a)=fourBlockFuture (-κ) j k := by |
| simp only [w,scaleIncrementChain_append,zeroWeightChain_scale,constantWeightChain_scale,neg_div]; rfl |
| have heFull : (b++l)++(j++k)++t=scaleIncrementChain β |
| (weightedUnderlying (C.c a)++weightedUnderlying (C.w a)++C.t) := by |
| simp only [C.c_underlying,C.w_underlying,scaleIncrementChain_append]; rfl |
| have hm : ∀ p∈(b++l)++(j++k)++t,p.1∈Icc (0:ℝ) 1 := by |
| rw [heFull]; exact scaleIncrementChain_mass β _ (C.full_mass a) |
| have hs : ((b++l)++(j++k)++t).Pairwise (fun p q => p.1≤q.1) := by |
| rw [heFull]; exact scaleIncrementChain_sorted β _ (C.full_sorted hmα a) |
| have hmF := fourBlockTent_mass κ b l j k (fun p hp => hm p (List.mem_append_left _ hp)) |
| have hsF := fourBlockTent_sorted κ b l j k (List.pairwise_append.mp hs).1 |
| have M := C.v_mean_bounds hmα hh hθ |
| have hM : 0≤weightedMean (scaleIncrementChain β (C.v θ)) := by |
| rw [weightedMean_scale] |
| exact mul_nonneg (sq_nonneg β) ((mul_nonneg hθ (hα r).1).trans M.1) |
| have hmean : 0≤weightedMean (fourBlockFuture η j k) := hv ▸ hM |
| have hB : 0≤β^2*a := mul_nonneg (sq_nonneg β) ha.le |
| have hn := ne_of_gt hh |
| have an := ne_of_gt ha |
| have hld : rawVariance l=(β^2*a)*(h/a) := by |
| dsimp only [l]; rw [rawVariance_scale,C.l_variance] |
| field_simp |
| have hjd : rawVariance j=(β^2*a)*(h/a) := by |
| dsimp only [j]; rw [rawVariance_scale,C.j_variance] |
| field_simp |
| have HK : κ*(h/a)=1 := by dsimp only [κ]; field_simp |
| have H := fourBlockQuadratic_tent_le κ η b l j k t hm hs hmean (div_pos hh ha) hB HK hld hjd |
| let A := scalarTentAtomMass (fourBlockTent κ b l j k).length (fun i => ((fourBlockTent κ b l j k).get i).1) |
| (fourBlockLeft κ b l j k) (fourBlockRight κ b l j k) |
| have hA : 0≤A := scalarTentAtomMass_nonneg _ _ (fun i => hmF _ (List.get_mem _ i)) |
| (mass_get_monotone hsF) _ _ |
| have hlL : 0<l.length := by |
| dsimp only [l]; simp only [scaleIncrementChain,List.length_map] |
| exact rawVariance_pos_length (show 0<rawVariance C.l by rw [C.l_variance]; exact hh) |
| have hjL : 0<j.length := by |
| dsimp only [j]; simp only [scaleIncrementChain,List.length_map] |
| exact rawVariance_pos_length (show 0<rawVariance C.j by rw [C.j_variance]; exact hh) |
| have hAω : A≤α (r+h)-α (r-h) := fourBlockTent_atom_le κ b l j k hlL hjL |
| (scaleIncrementChain_mass (P:=fun x => α (r-h)≤x) β _ (fun p hp => (C.left.raw_mass_interval hmα p hp).1)) |
| (scaleIncrementChain_mass (P:=fun x => x≤α (r+h)) β _ (fun p hp => (C.right.raw_mass_interval hmα p hp).2)) |
| have Herr := narrowTentError_mono (R:=weightedAbsCross (fourBlockFuture η j k)) hmean hB hA hAω |
| have H0 : narrowBaseQuadratic (fourBlockCommon κ b l) (fourBlockFuture η j k) (fourBlockFuture (-κ) j k) t≤narrowVariance (fourBlockCommon κ b l) (fourBlockFuture η j k) (fourBlockFuture (-κ) j k)* |
| narrowSusceptibility (fourBlockCommon κ b l) (fourBlockFuture (-κ) j k) t+ |
| 2*weightedMean (fourBlockFuture η j k)*(β^2*a)*narrowJointMoment (fourBlockCommon κ b l) (fourBlockFuture (-κ) j k) t+ |
| narrowTentError (weightedMean (fourBlockFuture η j k)) (β^2*a) (weightedAbsCross (fourBlockFuture η j k)) A := by |
| simpa only [narrowTentError,A,add_assoc] using H |
| have H' := H0.trans (add_le_add le_rfl Herr) |
| rw [← hc,← hv,← hw] at H' |
| rw [narrowVariance_scale,C.variance_stats hn,C.scalar_susceptibility β hα hmα ⟨hr.trans hrq,hq⟩ a, |
| C.scalar_joint β hα hmα hr hrq hq a,C.scaled_v_mean β hmα θ,weightedAbsCross_scale,C.v_absCross hh hθ] at H' |
| exact H' |
|
|
| end NarrowTimeBlocks |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
