| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.SqTwiceSqAbs |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| open scoped BigOperators ComplexConjugate |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Real Complex |
| open scoped ENNReal ComplexConjugate |
| open ProbabilityTheory |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology |
| open MeasureTheory Set |
| open scoped ENNReal |
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped NNReal |
|
|
| lemma ae_self_gradient_square {κ : Type*} [Fintype κ] {C : ℝ≥0} |
| {f : EuclideanSpace ℝ κ→ℝ} (hf : LipschitzWith C f) : |
| ∀ᵐ x ∂volume, DifferentiableAt ℝ (fun y ↦ f y^2) x ∧ |
| (∑ p, (fderiv ℝ (fun y ↦ f y^2) x (EuclideanSpace.basisFun κ ℝ p))^2)≤ |
| (4*(C:ℝ)^2)*f x^2 := by |
| filter_upwards [hf.ae_differentiableAt (μ:=volume)] with x hx |
| refine ⟨hx.pow 2,?_⟩ |
| rw [←(EuclideanSpace.basisFun κ ℝ).norm_dual,fderiv_fun_pow 2 hx] |
| simp only [Nat.reduceSub,pow_one,nsmul_eq_mul,Nat.cast_ofNat,norm_smul,Real.norm_eq_abs] |
| have hb := norm_fderiv_le_of_lipschitz ℝ (x₀:=x) hf |
| have hsq := (sq_le_sq₀ (norm_nonneg _) C.coe_nonneg).mpr hb |
| calc |
| _ = (4*f x^2)*‖fderiv ℝ f x‖^2 := by rw [mul_pow,sq_abs]; ring |
| _ ≤ (4*f x^2)*(C:ℝ)^2 := mul_le_mul_of_nonneg_left hsq (by positivity) |
| _ = _ := by ring |
|
|
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
|
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| theorem localCloseSmoothCount_ae_gradient (L : Length) {s : ℝ} (hs : 0<s) (hs1 : s≤1) (v : Torus L) : |
| ∀ᵐ x : EuclideanConfiguration ι ∂volume, |
| DifferentiableAt ℝ (fun y ↦ localCloseSmoothCount L s v (euclideanProject L y)) x ∧ |
| (∑ p, (fderiv ℝ (fun y ↦ localCloseSmoothCount L s v (euclideanProject L y)) x |
| (EuclideanSpace.basisFun (ι×Fin 3) ℝ p))^2)≤ |
| (4096/s^2)*localCloseSmoothCount L s v (euclideanProject L x) := by |
| have h := ae_self_gradient_square (localCloseRoot_euclidean_lipschitz (ι:=ι) L hs hs1 v) |
| simp_rw [localCloseRoot_sq] at h |
| have hc : (4:ℝ)*(32/s)^2=4096/s^2 := by ring |
| rw [←hc] |
| exact h |
|
|
| end HardSphere |
|
|
| end |
|
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| end OAI |
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|
