| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFSuffixSemigroup |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def standardGaussianAbsMoment : ℝ := ∫ z : ℝ, |z| ∂gaussianReal 0 1 |
|
|
| theorem standardGaussianAbsMoment_nonneg : 0 ≤ standardGaussianAbsMoment := |
| integral_nonneg (fun _ => abs_nonneg _) |
|
|
| theorem scalarStep_zero_mass_close {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {K : ℝ≥0} (hK : LipschitzWith K f) (v x : ℝ) : |
| |scalarStep 0 v f x-f x| ≤ (K:ℝ)*|v| *standardGaussianAbsMoment := by |
| let L : ℝ →L[ℝ] ℝ := v • ContinuousLinearMap.id ℝ ℝ |
| have hf' : Integrable (fun z : ℝ => f (x+v*z)) (gaussianReal 0 1) := |
| ((hf.translate x).compCLM L).hasExpGrowth.integrable_gaussianMeasure |
| (((hf.translate x).compCLM L).1.continuous) |
| have hi : Integrable (fun z : ℝ => |z|) (gaussianReal 0 1) := by |
| simpa only [Real.norm_eq_abs] using |
| (HasExpGrowth.norm_id (E:=ℝ)).integrable_gaussianMeasure continuous_norm (μ:=gaussianReal 0 1) |
| simp only [scalarStep,gaussianStep,ite_true] |
| have he : (∫ z : ℝ, f (x+v*z) ∂gaussianReal 0 1)-f x = |
| ∫ z : ℝ, f (x+v*z)-f x ∂gaussianReal 0 1 := by |
| rw [integral_sub hf' (integrable_const _)]; simp |
| rw [he] |
| calc |
| _ ≤ ∫ z : ℝ, |f (x+v*z)-f x| ∂gaussianReal 0 1 := by simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun z : ℝ => f (x+v*z)-f x) (μ:=gaussianReal 0 1) |
| _ ≤ ∫ z : ℝ, (K:ℝ)*|v| *|z| ∂gaussianReal 0 1 := by |
| apply integral_mono ((hf'.sub (integrable_const _)).norm) (hi.const_mul _) |
| intro z |
| have H := hK.dist_le_mul (x+v*z) x |
| simpa only [Real.dist_eq,Real.norm_eq_abs,Pi.sub_apply,add_sub_cancel_left,abs_mul,mul_assoc] using H |
| _ = _ := by rw [integral_const_mul]; rfl |
|
|
| theorem scalarCDFOperator_eq_zero_mass {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {K : ℝ≥0} (hK : LipschitzWith K f) (β : ℝ) {α : ℝ → ℝ} |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hm : Monotone α) (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) |
| (hzero : ∀ z∈Ico s (s+t), α z=0) : |
| scalarCDFOperator β α s t f=scalarTimeStep β 0 t f := by |
| funext x |
| exact scalarCDFOperator_eq_chain hf hK β ha hm [(0,t)] |
| (by simp) s t ht (by simp [chainDuration]) ⟨hzero,trivial⟩ x |
|
|
| theorem scalarCDFOperator_small_time {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| {K : ℝ≥0} (hK : LipschitzWith K f) (β : ℝ) {α : ℝ → ℝ} |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hm : Monotone α) (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) (x : ℝ) : |
| |scalarCDFOperator β α s t f x-f x| ≤ |
| scalarTimeMassConstantK β K*(t:ℝ)+(K:ℝ)*|β| *Real.sqrt (t:ℝ)*standardGaussianAbsMoment := by |
| have h0 : ∀ z : ℝ, (0:ℝ)∈Icc (0:ℝ) 1 := by simp |
| have H := scalarCDFOperator_cdf_lipschitz hf hK β ha hm h0 monotone_const s t ht x |
| rw [scalarCDFOperator_eq_zero_mass hf hK β h0 monotone_const s t ht (by simp)] at H |
| have hi : (∫ z in s..s+t, |α z-(0:ℝ)|) ≤ (t:ℝ) := by |
| have hi' := intervalIntegral.integral_mono_on (show s ≤ s+(t:ℝ) from le_add_of_nonneg_right t.coe_nonneg) |
| (hm.intervalIntegrable (μ:=volume)).abs (intervalIntegrable_const (c:=(1:ℝ))) |
| (fun z _ => by simpa only [sub_zero,abs_of_nonneg (ha z).1] using (ha z).2) |
| simpa only [sub_zero,intervalIntegral.integral_const,smul_eq_mul,add_sub_cancel_left,mul_one] using hi' |
| have H' := scalarStep_zero_mass_close hf hK (β*Real.sqrt (t:ℝ)) x |
| have he : |β*Real.sqrt (t:ℝ)|=|β| *Real.sqrt (t:ℝ) := by rw [abs_mul,abs_of_nonneg (Real.sqrt_nonneg _)] |
| rw [he] at H' |
| exact (abs_sub_le _ _ _).trans ((add_le_add (H.trans (mul_le_mul_of_nonneg_left hi |
| (scalarTimeMassConstantK_nonneg β K))) H').trans_eq (by ring)) |
|
|
| theorem scalarCDFValue_time_modulus (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {s q : ℝ} |
| (hs : 0 ≤ s) (hsq : s ≤ q) (hq : q ≤ 1) (x : ℝ) : |
| |scalarCDFValue β α s (Real.toNNReal (1-s)) x- |
| scalarCDFValue β α q (Real.toNNReal (1-q)) x| ≤ |
| scalarTimeMassConstantK β 1*(q-s)+|β| *Real.sqrt (q-s)*standardGaussianAbsMoment := by |
| have hr : Real.toNNReal (1-q) ≤ 1 := by |
| rw [Real.toNNReal_le_iff_le_coe,NNReal.coe_one]; linarith |
| have ht : Real.toNNReal (q-s) ≤ 1 := by |
| rw [Real.toNNReal_le_iff_le_coe,NNReal.coe_one]; linarith |
| rw [scalarCDFValue_suffix_semigroup β α ha h1 hs hsq hq] |
| have H := scalarCDFOperator_small_time (scalarCDFValue_regular β ha α.mono q _ hr) |
| (scalarCDFValue_lipschitz β ha α.mono q _ hr) β ha α.mono s _ ht x |
| simpa only [Real.coe_toNNReal _ (sub_nonneg.mpr hsq),NNReal.coe_one,one_mul] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
