| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.GroundPositionLaw |
| import OAI.Analysis.HardSphere.Geometry.SpatialScalarSquare |
| import OAI.Analysis.HardSphere.Geometry.MollifiedCloseJoint |
|
|
| 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 ComplexConjugate |
|
|
| lemma finset_erase_recursion_on {S : Type*} [DecidableEq S] (a : Finset S → ℝ) |
| (P : Finset S → Prop) (hP : ∀ A s, P A → P (A.erase s)) (q : ℝ) |
| (hq : 0≤q) (h0 : a ∅≤1) |
| (hr : ∀ A : Finset S, P A → A.Nonempty → (A.card:ℝ)*a A≤q*∑ s∈A, a (A.erase s)) : |
| ∀ A : Finset S, P A → a A≤q^A.card := by |
| intro A |
| induction A using Finset.strongInductionOn with |
| | _ A ih => |
| intro hPA |
| by_cases hA : A.Nonempty |
| · have hc : 0<(A.card:ℝ) := by exact_mod_cast hA.card_pos |
| have hs : (∑ s∈A, a (A.erase s))≤(A.card:ℝ)*q^(A.card-1) := by |
| calc |
| _ ≤ ∑ _s∈A, q^(A.card-1) := Finset.sum_le_sum fun s hs ↦ by |
| have hh := ih (A.erase s) (Finset.erase_ssubset hs) (hP A s hPA) |
| rwa [Finset.card_erase_of_mem hs] at hh |
| _ = _ := by simp |
| have he : q*((A.card:ℝ)*q^(A.card-1))=(A.card:ℝ)*q^A.card := by |
| rw [show q^A.card=q^(A.card-1)*q by |
| conv_lhs => rw [←Nat.sub_add_cancel hA.card_pos] |
| rw [pow_succ]] |
| ring |
| have ht := (hr A hPA hA).trans ((mul_le_mul_of_nonneg_left hs hq).trans_eq he) |
| exact (mul_le_mul_iff_right₀ hc).mp (by simpa only [mul_comm (A.card:ℝ)] using ht) |
| · have he : A=∅ := Finset.not_nonempty_iff_eq_empty.mp hA |
| simpa [he] using h0 |
|
|
| variable {n : ℕ} [Nontrivial (Fin n)] |
|
|
| lemma mollifiedCloseJoint_empty_onScalar (L : Length) (s h : ℝ) |
| (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (u : ScalarL2 n L) : |
| (mollifiedCloseJoint L s φ h ∅).onScalar u=u := by |
| apply Lp.ext |
| filter_upwards [(mollifiedCloseJoint L s φ h ∅).coe_onScalar u] with X hX |
| simpa only [mollifiedCloseJoint_apply,Finset.prod_empty,one_mul] using hX |
|
|
| lemma groundPositionLaw_le_closeJoint (L : Length) {s h : ℝ} (hs : 0<s) (hs1 : s≤1) (hh : 0<h) |
| (φ : ContDiffBump (0 : EuclideanConfiguration (Fin n))) (hφ : ((32/s)*φ.rOut)^2≤h/4) |
| (A : Finset (Torus L)) (g : Jet n L) (T : Set (Configuration n L)) (hT : MeasurableSet T) |
| (hcount : ∀ X∈T, ∀ v∈A, 8*h≤((localCloseLabels L s v X).card:ℝ)) : |
| groundPositionLaw n L g T≤ENNReal.ofReal (‖(mollifiedCloseJoint L s φ h A).onScalar g.1‖^2) := by |
| rw [groundPositionLaw,withDensity_apply _ hT, |
| ←ofReal_integral_eq_lintegral_ofReal |
| ((Lp.memLp g.1).integrable_norm_pow (by norm_num : (2:ℕ)≠0)).integrableOn |
| (Eventually.of_forall fun X ↦ sq_nonneg ‖g.1 X‖),IndicatorStrong.norm_sq_integral] |
| apply ENNReal.ofReal_le_ofReal |
| have hi : Integrable (fun X ↦ ‖((mollifiedCloseJoint L s φ h A).onScalar g.1) X‖^2) volume := |
| (Lp.memLp _).integrable_norm_pow (by norm_num) |
| calc |
| _ = ∫ X in T, ‖((mollifiedCloseJoint L s φ h A).onScalar g.1) X‖^2 ∂volume := by |
| apply setIntegral_congr_ae hT |
| filter_upwards [(mollifiedCloseJoint L s φ h A).coe_onScalar g.1] with X hX hXT |
| rw [hX,mollifiedCloseJoint_covers L hs hs1 hh φ hφ A X (hcount X hXT),one_mul] |
| _ ≤ _ := setIntegral_le_integral hi (ae_of_all _ fun X ↦ sq_nonneg _) |
|
|
| end HardSphere |
|
|
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| lemma close_support_integral {n : ℕ} {L : Length} (A : Finset (Torus L)) |
| {s C c b D h : ℝ} (hc : 0<c) (hb : 0<b) |
| (hcoef : (C+1)*D≤c/b^3*(h/4)) |
| (f : SmoothMultiplier n L) (g : ScalarL2 n L) |
| (hsupport : ∀ X, f.toFun X≠0 → (h/4)*(A.card:ℝ)≤((localizedNearLabels L A s X).card:ℝ)) : |
| D*(A.card:ℝ)*‖f.onScalar g‖^2≤ |
| ∫ X, (weightedSiteCount L A X-C*D*(A.card:ℝ)+ |
| c/b^3*((localizedNearLabels L A s X).card:ℝ))*‖(f.onScalar g) X‖^2 := by |
| have hn : Integrable (fun X ↦ ‖(f.onScalar g) X‖^2) volume := |
| (Lp.memLp _).integrable_norm_pow (by norm_num) |
| have hf := weightedSiteCount_integrable L A (f.onScalar g) |
| have hk := localizedNearLabels_integrable L A s (f.onScalar g) |
| have hi : Integrable (fun X ↦ (weightedSiteCount L A X-C*D*(A.card:ℝ)+ |
| c/b^3*((localizedNearLabels L A s X).card:ℝ))*‖(f.onScalar g) X‖^2) volume := by |
| have he : (fun X ↦ (weightedSiteCount L A X-C*D*(A.card:ℝ)+ |
| c/b^3*((localizedNearLabels L A s X).card:ℝ))*‖(f.onScalar g) X‖^2) = |
| fun X ↦ (weightedSiteCount L A X*‖(f.onScalar g) X‖^2- |
| (C*D*(A.card:ℝ))*‖(f.onScalar g) X‖^2)+ |
| (c/b^3)*(((localizedNearLabels L A s X).card:ℝ)*‖(f.onScalar g) X‖^2) := by |
| funext X |
| ring |
| rw [he] |
| exact (hf.sub (hn.const_mul (C*D*(A.card:ℝ)))).add (hk.const_mul (c/b^3)) |
| rw [IndicatorStrong.norm_sq_integral,←integral_const_mul] |
| apply integral_mono_ae (hn.const_mul _) hi |
| filter_upwards [f.coe_onScalar g] with X hX |
| by_cases hz : f.toFun X=0 |
| · simp only [hX,hz,zero_mul,norm_zero,zero_pow (by omega : (2:ℕ)≠0),mul_zero,le_rfl] |
| · have hkX := mul_le_mul_of_nonneg_left (hsupport X hz) (by positivity : 0≤c/b^3) |
| have hcX := mul_le_mul_of_nonneg_right hcoef (Nat.cast_nonneg A.card : (0:ℝ)≤A.card) |
| have hw := weightedSiteCount_nonneg L A X |
| apply mul_le_mul_of_nonneg_right _ (sq_nonneg _) |
| nlinarith only [hkX,hcX,hw] |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
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|
