| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarCDFLimit |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchySandwich |
| import OAI.Probability.SKBarriers.Hierarchy.HierarchyScoreGeneral |
| import OAI.Probability.SKBarriers.Scalar.ScalarSymmetricTaylor |
| import OAI.Probability.SKBarriers.Scalar.UniformExpansion |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| section Linear |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
|
|
| theorem scalar_composition_coordinate_bound {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| (hL : LipschitzWith 1 f) (X : E →L[ℝ] ℝ) (z e : E) : |
| |fderiv ℝ (f ∘ X) z e| ≤ |X e| := by |
| rw [fderiv_comp z (hf.1.differentiable (by norm_num) _) X.differentiableAt, |
| X.fderiv,ContinuousLinearMap.comp_apply] |
| have H := ((fderiv ℝ f (X z)).le_opNorm (X e)).trans |
| (mul_le_mul_of_nonneg_right (norm_fderiv_le_of_lipschitz ℝ hL) (norm_nonneg _)) |
| simpa only [Real.norm_eq_abs,NNReal.coe_one,one_mul] using H |
| end Linear |
|
|
| def tripleMiddleBoundary (f : ℝ → ℝ) {n : ℕ} |
| (X Z : ParameterSpace n →L[ℝ] ℝ) : ParameterSpace n → ℝ := |
| fun z => f (X z)+2*f (Z z) |
|
|
| def tripleModifiedBoundary (f : ℝ → ℝ) {n : ℕ} |
| (X Z Y : ParameterSpace n →L[ℝ] ℝ) (δ : ℝ) : ParameterSpace n → ℝ := |
| fun z => f (X z)+f (Z z+δ*Y z)+f (Z z-δ*Y z) |
|
|
| theorem tripleMiddleBoundary_regular {n : ℕ} {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| (X Z : ParameterSpace n →L[ℝ] ℝ) : BoundedDerivs (tripleMiddleBoundary f X Z) := |
| (hf.compCLM X).add ((hf.compCLM Z).const_mul 2) |
|
|
| theorem tripleModifiedBoundary_regular {n : ℕ} {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| (X Z Y : ParameterSpace n →L[ℝ] ℝ) (δ : ℝ) : |
| BoundedDerivs (tripleModifiedBoundary f X Z Y δ) := |
| ((hf.compCLM X).add (hf.compCLM (Z+δ • Y))).add (hf.compCLM (Z-δ • Y)) |
|
|
| theorem tripleMiddleBoundary_coordinate_bound {n : ℕ} {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| (hL : LipschitzWith 1 f) (X Z : ParameterSpace n →L[ℝ] ℝ) (i : Fin n) (z : ParameterSpace n) : |
| |fderiv ℝ (tripleMiddleBoundary f X Z) z (coordinateAxis n i)| ≤ |
| |X (coordinateAxis n i)|+2*|Z (coordinateAxis n i)| := by |
| let F := f ∘ X |
| let G := f ∘ Z |
| have hF := (hf.compCLM X).1.differentiable (by norm_num) |
| have hG := (hf.compCLM Z).1.differentiable (by norm_num) |
| change |fderiv ℝ (fun z => F z+2*G z) z (coordinateAxis n i)| ≤ _ |
| dsimp only [F,G,Function.comp_def] |
| rw [fderiv_fun_add (hF z) ((hG z).const_mul 2),fderiv_const_mul (hG z), |
| add_apply,smul_apply,smul_eq_mul] |
| calc |
| _ ≤ |fderiv ℝ F z (coordinateAxis n i)|+|2*fderiv ℝ G z (coordinateAxis n i)| := abs_add_le _ _ |
| _ ≤ _ := by |
| rw [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] |
| exact add_le_add (scalar_composition_coordinate_bound hf hL X z _) |
| (mul_le_mul_of_nonneg_left (scalar_composition_coordinate_bound hf hL Z z _) (by norm_num)) |
|
|
| theorem tripleMiddle_recursive_expansion (n t : ℕ) (m : Fin n → ℝ) (p v : Fin t → ℝ) |
| (hm : ∀ i, m i∈Icc (0:ℝ) 1) (hmono : Monotone m) |
| (hp : ∀ i, p i∈Icc (0:ℝ) 1) (hpmono : Monotone p) |
| (X Z : ParameterSpace n →L[ℝ] ℝ) (a : Fin n → ℝ) (x : ℝ) |
| {c δ : ℝ} (hc : 0<c) (hδ : 4*δ^2 ≤ c) (hδ1 : |δ| ≤ 1) |
| (hsmall : c*(4*(∑ i, (a i)^2)+ |
| (2*∑ i, |a i| *(|X (coordinateAxis n i)|+2*|Z (coordinateAxis n i)|))^2) ≤ 1/2) : |
| let f := scalarHierarchy t p v scalarSpinTerminal |
| let F := tripleMiddleBoundary f X Z |
| let Y := coordinateLinear n a |
| hierarchyPressure n m (tripleModifiedBoundary f X Z Y δ) x ≤ |
| hierarchyPressure n m F x+ |
| δ^2*(∫ z, rootHessian 0 f (Z z)*(Y z)^2 ∂hierarchyPathLaw n m F x)+ |
| ((2*(susceptibilityLipschitzConstant:ℝ)*(1+2/c^2)+32/c^2)*2)*|δ|^3 := by |
| dsimp only |
| let f := scalarHierarchy t p v scalarSpinTerminal |
| let F := tripleMiddleBoundary f X Z |
| let Y := coordinateLinear n a |
| let G := tripleModifiedBoundary f X Z Y δ |
| let μ := hierarchyPathLaw n m F x |
| have hf : BoundedDerivs f := scalarHierarchy_spin_regular t p v |
| have hFL : LipschitzWith 1 f := scalarHierarchy_lipschitz t p v (fun i => (hp i).1) |
| scalarSpinTerminal_regular scalarSpinTerminal_lipschitz |
| have hF : BoundedDerivs F := tripleMiddleBoundary_regular hf X Z |
| have hG : BoundedDerivs G := tripleModifiedBoundary_regular hf X Z Y δ |
| have hU : BoundedDerivs (fun z => G z-F z) := by |
| simpa only [neg_one_mul,← sub_eq_add_neg] using hG.add (hF.const_mul (-1)) |
| let := hierarchyPathLaw_probability n m F hF x |
| obtain ⟨hE,hM⟩ := hierarchy_exp_square_bound n m hm hmono F hF |
| (fun i => |X (coordinateAxis n i)|+2*|Z (coordinateAxis n i)|) (by intro i; positivity) |
| (fun z i => tripleMiddleBoundary_coordinate_bound hf hFL X Z i z) x a hc.le hsmall |
| have hΔ : Integrable (fun z => G z-F z) μ := |
| hU.hasExpGrowth.integrable_hierarchyPathLaw n m hF hU.1.continuous x |
| have hcA : Continuous (fun z : ParameterSpace n => rootHessian 0 f (Z z)) := |
| (scalarHierarchy_spin_hessian_lipschitz t p v hp hpmono).continuous.comp Z.continuous |
| have hbA (z : ParameterSpace n) : ‖rootHessian 0 f (Z z)‖ ≤ 1 := by |
| have H := scalarHierarchy_spin_derivative_bounds t p v hp (Z z) |
| change |fderiv ℝ (fderiv ℝ f) (Z z) 1 1| ≤ 1 |
| rw [abs_of_pos H.2.1] |
| exact H.2.2.trans (sub_le_self _ (sq_nonneg _)) |
| have hA : Integrable (fun z => rootHessian 0 f (Z z)*(Y z)^2) μ := |
| ((HasExpGrowth.of_bounded zero_le_one hbA).mul ((HasExpGrowth.linear Y).pow 2)).integrable_hierarchyPathLaw |
| n m hF (hcA.mul (Y.continuous.pow 2)) x |
| have he (z) : G z-F z=symmetricIncrement f (Z z) (δ*Y z) := by |
| dsimp [F,G,tripleMiddleBoundary,tripleModifiedBoundary,symmetricIncrement]; ring |
| have H := log_integral_exp_cubic_expansion μ Y (fun z => G z-F z) |
| (fun z => rootHessian 0 f (Z z)) Y.continuous.aestronglyMeasurable hΔ hA hc hδ hδ1 |
| (by positivity : 0 ≤ 2*(susceptibilityLipschitzConstant:ℝ)) hE hM |
| (ae_of_all _ (fun z => by rw [he]; exact (scalarHierarchy_symmetricIncrement_bounds t p v hp hpmono _ δ _).1)) |
| (ae_of_all _ (fun z => by rw [he]; exact (scalarHierarchy_symmetricIncrement_bounds t p v hp hpmono _ δ _).2)) |
| have HS := (hierarchyPressure_increment_integral_sandwich n m hm hF hG x).2 |
| change hierarchyPressure n m G x ≤ hierarchyPressure n m F x+_+_ |
| linarith |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
