| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.CenteredCircleLift |
|
|
| 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 |
|
|
| lemma finite_fiber_sum_le {ι : Type*} [Fintype ι] [DecidableEq ι] |
| (f : ι→ι) (a : ι→ℝ) (C : ℕ) (ha : ∀ i, 0≤a i) |
| (hc : ∀ k, (Finset.univ.filter (fun i ↦ f i=k)).card≤C) : |
| (∑ i, a (f i))≤(C:ℝ)*∑ i, a i := by |
| rw [←Finset.sum_fiberwise Finset.univ f (fun i ↦ a (f i)),Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro k _ |
| calc |
| _ = ∑ i∈Finset.univ.filter (fun i ↦ f i=k), a k := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [(Finset.mem_filter.mp hi).2] |
| _ = ((Finset.univ.filter (fun i ↦ f i=k)).card:ℝ)*a k := by simp |
| _ ≤ _ := mul_le_mul_of_nonneg_right (by exact_mod_cast hc k) (ha k) |
|
|
| theorem nearestParticleDistance_sum_sq {ι : Type*} [Fintype ι] [DecidableEq ι] [Nontrivial ι] |
| (L : Length) (X Y : ι→Torus L) (hX : Function.Injective X) (hY : Function.Injective Y) : |
| (∑ i, (nearestParticleDistance L Y i-nearestParticleDistance L X i)^2)≤ |
| 502*∑ i, torusDistance L (X i) (Y i)^2 := by |
| let δ : ι→ℝ := fun i ↦ torusDistance L (X i) (Y i) |
| have hδ (i : ι) : 0≤δ i := Real.sqrt_nonneg _ |
| have hpoint (i : ι) : (nearestParticleDistance L Y i-nearestParticleDistance L X i)^2≤ |
| 2*(δ i)^2+2*(δ (nearestLabel L X i))^2+2*(δ (nearestLabel L Y i))^2 := by |
| have hp := nearestParticleDistance_sub_le L X Y i |
| have hm := nearestParticleDistance_sub_le L Y X i |
| have hs (j : ι) : torusDistance L (Y j) (X j)=δ j := by simp only [δ,torusDistance,dist_comm] |
| change nearestParticleDistance L Y i-nearestParticleDistance L X i≤δ i+δ (nearestLabel L X i) at hp |
| simp only [hs] at hm |
| by_cases h : 0≤nearestParticleDistance L Y i-nearestParticleDistance L X i |
| · have hsq := (sq_le_sq₀ h (add_nonneg (hδ i) (hδ _))).mpr hp |
| nlinarith only [hsq,sq_nonneg (δ i-δ (nearestLabel L X i)),sq_nonneg (δ (nearestLabel L Y i))] |
| · have h' : 0≤nearestParticleDistance L X i-nearestParticleDistance L Y i := by linarith only [h] |
| have hsq := (sq_le_sq₀ h' (add_nonneg (hδ i) (hδ _))).mpr hm |
| nlinarith only [hsq,sq_nonneg (δ i-δ (nearestLabel L Y i)),sq_nonneg (δ (nearestLabel L X i))] |
| have hSX := finite_fiber_sum_le (nearestLabel L X) (fun i ↦ (δ i)^2) 125 |
| (fun _ ↦ sq_nonneg _) (nearestLabel_fiber_card L X hX) |
| have hSY := finite_fiber_sum_le (nearestLabel L Y) (fun i ↦ (δ i)^2) 125 |
| (fun _ ↦ sq_nonneg _) (nearestLabel_fiber_card L Y hY) |
| calc |
| _ ≤ ∑ i, (2*(δ i)^2+2*(δ (nearestLabel L X i))^2+2*(δ (nearestLabel L Y i))^2) := |
| Finset.sum_le_sum (fun i _ ↦ hpoint i) |
| _ = 2*(∑ i, (δ i)^2)+2*(∑ i, (δ (nearestLabel L X i))^2)+2*(∑ i, (δ (nearestLabel L Y i))^2) := by |
| simp only [Finset.sum_add_distrib,Finset.mul_sum] |
| _ ≤ 502*∑ i, (δ i)^2 := by norm_num at hSX hSY; linarith only [hSX,hSY] |
|
|
| end HardSphere |
|
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| end |
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| end OAI |
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