| import Mathlib |
| import OAI.Analysis.HardSphere.Orbitals.CenteredCell |
| import OAI.Analysis.HardSphere.Geometry.SelectedUnitCellPenalty |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal Topology |
| namespace HardSphere.CellOrbital |
|
|
| def gridHome {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (x : Torus L) : Fin 3→Fin K := |
| fun j ↦ unitIndex hL (x j+(1/2:ℝ)) |
|
|
| lemma gridHome_measurable {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) : Measurable (gridHome hL) := by |
| exact Measurable.of_eval (fun j ↦ (measurable_unitIndex hL).comp ((measurable_pi_apply j).add_const _)) |
|
|
| lemma gridHome_mem {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) (x : Torus L) : |
| x∈centeredCell L (gridCenter L (gridHome hL x)) := by |
| apply (centeredCell_image (by rw [hL]; exact_mod_cast hK) _ _).mpr |
| intro j |
| let z := (AddCircle.equivIco L.val 0) (x j+(1/2:ℝ)) |
| have hz0 : 0≤z.val := z.property.1 |
| let n : ℕ := ⌊z.val⌋₊ |
| refine ⟨z.val-(n:ℝ)-1/2,⟨?_,?_⟩,?_⟩ |
| · linarith [Nat.floor_le hz0] |
| · have hh := Nat.lt_floor_add_one z.val |
| dsimp [n] |
| linarith |
| · have he : (z.val : AddCircle L.val)=x j+((1/2:ℝ):AddCircle L.val) := AddCircle.coe_equivIco |
| change ((z.val-(n:ℝ)-1/2:ℝ):AddCircle L.val)=x j-((n:ℝ):AddCircle L.val) |
| simp only [QuotientAddGroup.mk_sub] |
| rw [he] |
| abel |
|
|
| lemma centeredCell_disjoint {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| {v w : Fin 3→Fin K} (hvw : v≠w) : Disjoint (centeredCell L (gridCenter L v)) (centeredCell L (gridCenter L w)) := by |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| obtain ⟨j,hj⟩ : ∃j,v j≠w j := by |
| by_contra h |
| push Not at h |
| exact hvw (funext h) |
| have h1 : 1≤L.val := by rw [hL]; exact_mod_cast hK |
| obtain ⟨s,hs,hes⟩ := (centeredCell_image h1 _ _).mp hx j |
| obtain ⟨t,ht,het⟩ := (centeredCell_image h1 _ _).mp hy j |
| have he : gridCenter L v j-gridCenter L w j=((t-s:ℝ):AddCircle L.val) := by |
| rw [QuotientAddGroup.mk_sub,het,hes] |
| abel |
| have ha : |t-s|<1 := by rw [abs_lt]; constructor <;> linarith [ht.1,ht.2,hs.1,hs.2] |
| have hd : dist (gridCenter L v j) (gridCenter L w j)<1 := by |
| rw [dist_eq_norm,he] |
| exact lt_of_le_of_lt (QuotientAddGroup.norm_mk_le_norm (m:=t-s)) (by simpa only [Real.norm_eq_abs] using ha) |
| exact (not_lt_of_ge (fin_circle_dist hL hj)) hd |
|
|
| lemma gridHome_eq_iff {K : ℕ} {L : Length} (hL : L.val=(K:ℝ)) (hK : 1≤K) |
| (x : Torus L) (v : Fin 3→Fin K) : gridHome hL x=v ↔ x∈centeredCell L (gridCenter L v) := by |
| constructor |
| · intro h; exact h ▸ gridHome_mem hL hK x |
| · intro hx |
| by_contra h |
| exact Set.disjoint_left.mp (centeredCell_disjoint hL hK h) (gridHome_mem hL hK x) hx |
| |
| end HardSphere.CellOrbital |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
