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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.SqTwiceSqAbs
namespace OAI
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]
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
end OAI
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