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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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 | 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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