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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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.ParticleGradSq
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.SphericalCoordinates
open scoped BigOperators
noncomputable def activeLabels {α ι : Type*} [Fintype α] (σ : α → ι) : Finset α := by
classical
exact Finset.univ.filter (fun i ↦ ∃ j, j≠i ∧ σ j=σ i)
lemma mem_activeLabels {α ι : Type*} [Fintype α] (σ : α → ι) (i : α) :
i∈activeLabels σ ↔ ∃ j, j≠i ∧ σ j=σ i := by
classical
simp only [activeLabels,Finset.mem_filter,Finset.mem_univ,true_and]
lemma inactive_card_le {α ι : Type*} [Fintype α] [Fintype ι] [DecidableEq α] (σ : α → ι) :
(activeLabels σ)ᶜ.card≤Fintype.card ι := by
classical
apply Finset.card_le_card_of_injOn σ (by simp)
intro a ha b _ hab
by_contra hne
have ha' : a∉activeLabels σ := Finset.mem_compl.mp ha
apply ha'
simp only [activeLabels,Finset.mem_filter,Finset.mem_univ,true_and]
exact ⟨b,(fun h ↦ hne h.symm),hab.symm⟩
lemma active_card_lower {α ι : Type*} [Fintype α] [Fintype ι] (σ : α → ι) :
Fintype.card α-Fintype.card ι≤(activeLabels σ).card := by
classical
have h := inactive_card_le σ
rw [Finset.card_compl] at h
have hc := (activeLabels σ).card_le_univ
omega
lemma sector_capacity {n : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : (Fin (n+1) → Point) → E} (hF : ContDiff ℝ 1 F)
{r l : ℝ} (hr : 0<r) (hl : 0≤l) {ι : Type*} [Fintype ι]
(b : ι → Point) (σ : Fin (n+1) → ι)
(hz : ∀X a c, a≠c → sqRadius (X a-X c)≤r^2 → F X=0) :
((n+1-Fintype.card ι : ℕ) : ℝ≥0∞)*
(∫⁻ X, ENNReal.ofReal (‖F X‖^2) ∂Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l))) ≤
ENNReal.ofReal (Real.sqrt 3*l^3/r)*
∑ i, (∫⁻ X, ENNReal.ofReal (particleGradSq F i X)
∂Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l))) := by
classical
let μ := Measure.pi (fun a ↦ volume.restrict (cube (b (σ a)) l))
let M := ∫⁻ X, ENNReal.ofReal (‖F X‖^2) ∂μ
let G := fun i ↦ ∫⁻ X, ENNReal.ofReal (particleGradSq F i X) ∂μ
let C := ENNReal.ofReal (Real.sqrt 3*l^3/r)
have hi (i : Fin (n+1)) (hi : i∈activeLabels σ) : M≤C*G i := by
obtain ⟨j,hji,hs⟩ := (mem_activeLabels σ i).mp hi
obtain ⟨k,rfl⟩ := Fin.exists_succAbove_eq hji
exact sector_coordinate_capacity hF hr hl (fun a ↦ b (σ a)) i k (congrArg b hs) hz
have hs := Finset.sum_le_sum hi
have hcard := active_card_lower σ
rw [Fintype.card_fin] at hcard
calc
_ ≤ ((activeLabels σ).card : ℝ≥0∞)*M := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard) (by positivity)
_ = ∑ _ ∈ activeLabels σ, M := by simp
_ ≤ ∑ i ∈ activeLabels σ, C*G i := hs
_ = C*∑ i ∈ activeLabels σ, G i := (Finset.mul_sum _ _ _).symm
_ ≤ C*∑ i, G i := by gcongr; exact Finset.subset_univ _
_ = _ := rfl
end HardSphere.SphericalCoordinates
end
end OAI
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