| import OAI.Geometry.PeriodicTiling.SeparatedCenters |
| import Mathlib.Algebra.Order.Floor.Ring |
| import Mathlib.Data.Set.Countable |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree |
|
|
| open Set MeasureTheory |
|
|
| theorem AETiles.eq_of_voxel_close {d : ℕ} {F : Finset (Lattice d)} |
| {A : Set (Space d)} (h : AETiles (Thickening F) A) |
| {z : Lattice d} (hz : z ∈ F) {a b : A} |
| (hc : ∀ i, |(b : Space d) i - (a : Space d) i| < 1) : a = b := by |
| by_contra hne |
| have hpos := unitVoxel_inter_volume_pos |
| ((a : Space d) + castLattice z) ((b : Space d) + castLattice z) (by |
| intro i |
| simpa only [Pi.add_apply, add_sub_add_right_eq_sub] using hc i) |
| have hsub : ∀ c : A, unitVoxel ((c : Space d) + castLattice z) ⊆ |
| {x | x - (c : Space d) ∈ Thickening F} := by |
| intro c x hx |
| refine ⟨z, hz, ?_⟩ |
| simpa only [mem_unitVoxel, sub_add_eq_sub_sub] using hx |
| have hnull := (h.aedisjoint hne).mono (hsub a) (hsub b) |
| exact (ne_of_gt hpos) hnull |
|
|
| theorem AETiles.floor_injective {d : ℕ} {F : Finset (Lattice d)} |
| {A : Set (Space d)} (h : AETiles (Thickening F) A) (hF : F.Nonempty) : |
| Function.Injective (fun a : A => (fun i => ⌊(a : Space d) i⌋ : Lattice d)) := by |
| obtain ⟨z, hz⟩ := hF |
| intro a b he |
| apply h.eq_of_voxel_close hz |
| intro i |
| have hi : ⌊(a : Space d) i⌋ = ⌊(b : Space d) i⌋ := congrFun he i |
| have ha := Int.floor_le ((a : Space d) i) |
| have ha' := Int.lt_floor_add_one ((a : Space d) i) |
| have hb := Int.floor_le ((b : Space d) i) |
| have hb' := Int.lt_floor_add_one ((b : Space d) i) |
| rw [hi] at ha ha' |
| apply abs_lt.mpr |
| constructor <;> linarith |
|
|
| theorem AETiles.countable_of_thickening {d : ℕ} {F : Finset (Lattice d)} |
| {A : Set (Space d)} (h : AETiles (Thickening F) A) (hF : F.Nonempty) : |
| A.Countable := |
| Set.countable_coe_iff.mp (h.floor_injective hF).countable |
|
|
| end PeriodicTilingThree |
|
|
| end |
|
|
| end OAI |
|
|
