| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.SmoothMultiplier |
| import OAI.Analysis.HardSphere.Geometry.TorusLiftBox |
|
|
| 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 |
| namespace HardSphere |
| open MeasureTheory Set Filter |
| open scoped ComplexConjugate |
|
|
| def SmoothMultiplier.finsetProd {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} |
| (A : Finset S) (w : S → SmoothMultiplier n L) : SmoothMultiplier n L := |
| .ofSmooth (fun X ↦ ∏ s∈A, (w s).toFun X) (contDiff_prod fun s _ ↦ (w s).smooth) |
|
|
| lemma SmoothMultiplier.partial_finsetProd {S : Type*} [DecidableEq S] {n : ℕ} {L : Length} |
| (A : Finset S) (w : S → SmoothMultiplier n L) (p : Direction n) (X : Configuration n L) : |
| coordDerivative n L (SmoothMultiplier.finsetProd A w).toFun p X= |
| ∑ s∈A, (∏ z∈A.erase s, (w z).toFun X)*coordDerivative n L (w s).toFun p X := by |
| unfold coordDerivative |
| change (fderiv ℝ (fun Y ↦ ∏ s∈A, (w s).toFun (project n L Y)) _) _ = _ |
| rw [fderiv_finsetProd (fun s _ ↦ (w s).smooth.differentiable (by simp) _),sum_apply] |
| simp only [smul_apply,smul_eq_mul,project_representative] |
|
|
| lemma norm_sq_sum_of_disjoint {S : Type*} [DecidableEq S] (A : Finset S) (f : S → ℂ) |
| (hf : ∀ s∈A, ∀ z∈A, s≠z → f s=0 ∨ f z=0) : ‖∑ s∈A, f s‖^2=∑ s∈A, ‖f s‖^2 := by |
| by_cases h : ∃ s∈A, f s≠0 |
| · obtain ⟨s,hs,hfs⟩ := h |
| have hz : ∀ z∈A, z≠s → f z=0 := by |
| intro z hz hzs |
| exact (hf z hz s hs hzs).resolve_right hfs |
| rw [Finset.sum_eq_single s (fun z hza hzs ↦ hz z hza hzs) (by simp_all)] |
| rw [Finset.sum_eq_single s (fun z hza hzs ↦ by rw [hz z hza hzs,norm_zero]; norm_num) (by simp_all)] |
| · push Not at h |
| rw [Finset.sum_eq_zero (fun s hs ↦ h s hs)] |
| simp only [norm_zero,zero_pow (by norm_num : 2≠0)] |
| symm |
| exact Finset.sum_eq_zero fun s hs ↦ by rw [h s hs,norm_zero]; norm_num |
|
|
| end HardSphere |
|
|
| end |
|
|
| end OAI |
|
|
