| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.GridAlignment |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set |
| namespace HardSphere.Groups |
| open HardSphere.Lattice HardSphere.CellOrbital |
| attribute [local instance] Classical.propDecidable |
| lemma grid_near_of_distance {K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (v w : Fin 3→Fin K) (h : torusDistance L (gridCenter L v) (gridCenter L w)≤23) : |
| torusNear K 24 (gridSite w) (gridSite v) := by |
| obtain ⟨z,hz,he⟩ := grid_packing_witness L hL (fun j ↦ ((w j).val:ℝ)) v |
| (fun j ↦ (dist_coord_le_torusDistance L (gridCenter L v) (gridCenter L w) j).trans h) |
| refine ⟨fun j ↦ z j-(w j).val,?_,?_⟩ |
| · intro j |
| obtain ⟨hl,hu⟩ := Finset.mem_Icc.mp hz |
| have hl' := hl j |
| have hu' := hu j |
| simp only [Int.floor_natCast] at hl' hu' |
| change |z j-(w j).val|≤24 |
| exact abs_le.mpr ⟨by omega,by omega⟩ |
| · intro j |
| have hj := congrFun he j |
| dsimp [torusProjection,gridSite] at hj ⊢ |
| push_cast |
| rw [←hj] |
| ring |
| lemma distance_far_of_grid_far {K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| {R : ℕ} (hR : 24≤R) (v w : Fin 3→Fin K) |
| (h : ¬torusNear K R (gridSite v) (gridSite w)) : |
| 23<torusDistance L (gridCenter L v) (gridCenter L w) := by |
| by_contra hn |
| apply h |
| exact torusNear_mono hR (torusNear_symm (grid_near_of_distance L hL v w (le_of_not_gt hn))) |
|
|
| end HardSphere.Groups |
|
|
| end |
| end |
|
|
| end |
|
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| end OAI |
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|
