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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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