| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.EndpointsExterior |
| import OAI.Analysis.HardSphere.Allocation.PhysicalVertex |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
|
|
| def siteIndex {K : ℕ} [NeZero K] (v : DiscreteTorus K) : Fin 3→Fin K := |
| fun j ↦ ⟨(v j).val,ZMod.val_lt _⟩ |
|
|
| def indexSite {K : ℕ} (v : Fin 3→Fin K) : DiscreteTorus K := |
| fun j ↦ ((v j).val:ZMod K) |
|
|
| @[simp] lemma indexSite_siteIndex {K : ℕ} [NeZero K] (v : DiscreteTorus K) : |
| indexSite (siteIndex v)=v := by |
| funext j |
| exact ZMod.natCast_zmod_val _ |
|
|
| @[simp] lemma siteIndex_indexSite {K : ℕ} [NeZero K] (v : Fin 3→Fin K) : |
| siteIndex (indexSite v)=v := by |
| funext j |
| apply Fin.ext |
| change (((v j).val:ZMod K).val)=(v j).val |
| rw [ZMod.val_natCast,Nat.mod_eq_of_lt (v j).isLt] |
|
|
| def siteIndexEquiv {K : ℕ} [NeZero K] : DiscreteTorus K ≃ (Fin 3→Fin K) := |
| ⟨siteIndex,indexSite,indexSite_siteIndex,siteIndex_indexSite⟩ |
|
|
| lemma intPeriod_project {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) |
| (z : ℤ) (hz : (z:ZMod K)=0) : ((z:ℝ):AddCircle L.val)=0 := by |
| obtain ⟨q,hq⟩ := (ZMod.intCast_zmod_eq_zero_iff_dvd z K).mp hz |
| apply (AddCircle.coe_eq_zero_iff L.val).mpr |
| refine ⟨q,?_⟩ |
| rw [zsmul_eq_mul,hL] |
| norm_cast |
| rw [hq,mul_comm] |
|
|
| lemma siteIndex_project_lift {K : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) (z : Vertex) : |
| projectPoint L (anchorPoint (siteIndex (torusProjection K z)))= |
| projectPoint L (fun k ↦ (z k:ℝ)) := by |
| ext k |
| change ((((z k:ZMod K).val:ℕ):ℝ):AddCircle L.val)=((z k:ℝ):AddCircle L.val) |
| apply sub_eq_zero.mp |
| rw [←AddCircle.coe_sub] |
| have H := intPeriod_project hL (((z k:ZMod K).val:ℤ)-z k) (by |
| simp only [Int.cast_sub,Int.cast_natCast,ZMod.natCast_zmod_val,sub_self]) |
| simpa only [Int.cast_sub,Int.cast_natCast] using H |
|
|
| lemma targetStep_positive (i : Fin 3) : |
| targetStep ⟨2*i.val+1,by omega⟩=fun k ↦ if k=i then (1:ℝ) else 0 := by |
| funext k |
| fin_cases i <;> fin_cases k <;> norm_num [targetStep] |
|
|
| lemma targetStep_negative (i : Fin 3) : |
| targetStep ⟨2*i.val+2,by omega⟩=fun k ↦ if k=i then (-1:ℝ) else 0 := by |
| funext k |
| fin_cases i <;> fin_cases k <;> norm_num [targetStep] |
|
|
| lemma nearest_targetStep {x y : Vertex} (h : nearestStep x y) : |
| ∃ j : Fin 7,targetStep j=fun k ↦ ((y k-x k:ℤ):ℝ) := by |
| obtain ⟨i,hi,hr⟩ := h |
| rcases (abs_eq (by norm_num : (0:ℤ)≤1)).mp hi with hp|hn |
| · refine ⟨⟨2*i.val+2,by omega⟩,?_⟩ |
| rw [targetStep_negative] |
| funext k |
| by_cases hk : k=i |
| · subst k |
| simp only |
| have H : y i-x i=-1 := by omega |
| rw [H]; norm_num |
| · simp only [ite_eq_right hk,hr k hk,sub_self,Int.cast_zero] |
| · refine ⟨⟨2*i.val+1,by omega⟩,?_⟩ |
| rw [targetStep_positive] |
| funext k |
| by_cases hk : k=i |
| · subst k |
| simp only |
| have H : y i-x i=1 := by omega |
| rw [H]; norm_num |
| · simp only [ite_eq_right hk,hr k hk,sub_self,Int.cast_zero] |
|
|
| theorem adjacent_routeCellPeriod {K : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) {w v : DiscreteTorus K} (h : (torusNearestGraph K).Adj w v) : |
| ∃ j : Fin 7,projectPoint L (routeCellPeriod (siteIndex v) (siteIndex w) j)=0 := by |
| obtain ⟨_,x,y,hx,hy,hn⟩ := h |
| obtain ⟨j,hj⟩ := nearest_targetStep hn |
| refine ⟨j,?_⟩ |
| subst w v |
| have hx' := siteIndex_project_lift hL x |
| have hy' := siteIndex_project_lift hL y |
| ext k |
| change (((anchorPoint (siteIndex (torusProjection K y)) k- |
| (anchorPoint (siteIndex (torusProjection K x)) k+targetStep j k)):ℝ):AddCircle L.val)=0 |
| rw [AddCircle.coe_sub,AddCircle.coe_add,show targetStep j k=((y k-x k:ℤ):ℝ) from congrFun hj k] |
| have HX := congrFun hx' k |
| have HY := congrFun hy' k |
| change ((anchorPoint (siteIndex (torusProjection K x)) k:ℝ):AddCircle L.val)=((x k:ℝ):AddCircle L.val) at HX |
| change ((anchorPoint (siteIndex (torusProjection K y)) k:ℝ):AddCircle L.val)=((y k:ℝ):AddCircle L.val) at HY |
| rw [HX,HY,Int.cast_sub,AddCircle.coe_sub] |
| abel |
|
|
| lemma self_routeCellPeriod {K : ℕ} {L : Length} (v : Fin 3→Fin K) : |
| projectPoint L (routeCellPeriod v v 0)=0 := by |
| simp only [routeCellPeriod,targetStep,Fin.val_zero,↓reduceIte,add_zero,sub_self] |
| rfl |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace HardSphere.Groups |
| open Wiener Bridge Sampling Comparison GaussianPacking PathCutoff CellOrbital HardSphere.Lattice |
|
|
| lemma nearest_project_distance {L : Length} {x y : Vertex} (h : nearestStep x y) : |
| torusDistance L (projectPoint L (fun k ↦ (x k:ℝ))) |
| (projectPoint L (fun k ↦ (y k:ℝ)))≤1 := by |
| obtain ⟨j,hj,hh⟩ := h |
| have hc (i : Fin 3) : dist (((x i:ℝ):AddCircle L.val)) ((y i:ℝ):AddCircle L.val)≤ |
| if i=j then (1:ℝ) else 0 := by |
| rw [dist_eq_norm,←AddCircle.coe_sub] |
| refine (QuotientAddGroup.norm_mk_le_norm (m:=(x i:ℝ)-(y i:ℝ))).trans ?_ |
| rw [Real.norm_eq_abs] |
| by_cases hi : i=j |
| · subst i |
| simp only |
| exact_mod_cast le_of_eq hj |
| · simp only [ite_eq_right hi,hh i hi,sub_self,abs_zero,le_refl] |
| unfold torusDistance |
| apply (Real.sqrt_le_iff).mpr |
| refine ⟨by norm_num,?_⟩ |
| calc |
| _ ≤ ∑ i : Fin 3,(if i=j then (1:ℝ) else 0)^2 := Finset.sum_le_sum fun i _ ↦ |
| (sq_le_sq₀ dist_nonneg (by split_ifs <;> norm_num)).mpr (hc i) |
| _ = 1^2 := by simp |
|
|
| lemma adjacent_center_distance {K : ℕ} [NeZero K] {L : Length} |
| (hL : L.val=(K:ℝ)) {x y : DiscreteTorus K} (h : (torusNearestGraph K).Adj x y) : |
| torusDistance L (gridCenter L (siteIndex x)) (gridCenter L (siteIndex y))≤1 := by |
| obtain ⟨_,v,w,hv,hw,hn⟩ := h |
| subst x y |
| change torusDistance L (projectPoint L (anchorPoint (siteIndex (torusProjection K v)))) |
| (projectPoint L (anchorPoint (siteIndex (torusProjection K w))))≤1 |
| rw [siteIndex_project_lift hL,siteIndex_project_lift hL] |
| exact nearest_project_distance hn |
|
|
| end HardSphere.Groups |
| end |
| end |
|
|
| end OAI |
|
|
