| import OAI.Geometry.PeriodicTiling.EuclideanBasic |
| import Mathlib.Algebra.Order.Floor.Ring |
| import Mathlib.Analysis.Normed.Group.Bounded |
| import Mathlib.Data.Int.Interval |
| import Mathlib.Data.Pi.Interval |
| import Mathlib.Data.Set.Countable |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree |
|
|
| def centerBoxIndex {n : ℕ} (r : ℝ) (x : Space n) : Lattice n := |
| fun i => ⌊x i / r⌋ |
|
|
| theorem norm_sub_lt_of_centerBoxIndex_eq {n : ℕ} {r : ℝ} (hr : 0 < r) |
| {x y : Space n} (hxy : centerBoxIndex r x = centerBoxIndex r y) : |
| ‖x - y‖ < r := by |
| apply (pi_norm_lt_iff hr).mpr |
| intro i |
| have heq : ⌊x i / r⌋ = ⌊y i / r⌋ := congrFun hxy i |
| have hxl := Int.floor_le (x i / r) |
| have hxu := Int.lt_floor_add_one (x i / r) |
| have hyl := Int.floor_le (y i / r) |
| have hyu := Int.lt_floor_add_one (y i / r) |
| rw [← heq] at hyl hyu |
| have hxl' := (le_div_iff₀ hr).mp hxl |
| have hxu' := (div_lt_iff₀ hr).mp hxu |
| have hyl' := (le_div_iff₀ hr).mp hyl |
| have hyu' := (div_lt_iff₀ hr).mp hyu |
| simp only [Pi.sub_apply, Real.norm_eq_abs] |
| apply abs_lt.mpr |
| constructor <;> nlinarith |
|
|
| theorem centerBoxIndex_injOn {n : ℕ} {C : Set (Space n)} {r : ℝ} |
| (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖)) : |
| Set.InjOn (centerBoxIndex r) C := by |
| intro x hx y hy hxy |
| by_contra hne |
| exact (not_lt_of_ge (hsep hx hy hne)) (norm_sub_lt_of_centerBoxIndex_eq hr hxy) |
|
|
| theorem centers_countable {n : ℕ} {C : Set (Space n)} {r : ℝ} |
| (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖)) : C.Countable := by |
| apply Set.countable_of_injective_of_countable_image (centerBoxIndex_injOn hr hsep) |
| exact Set.to_countable _ |
|
|
| theorem centerBoxIndex_image_finite {n : ℕ} {r : ℝ} (hr : 0 < r) |
| {K : Set (Space n)} (hK : Bornology.IsBounded K) : |
| (centerBoxIndex r '' K).Finite := by |
| obtain ⟨R, hR⟩ := hK.exists_norm_le |
| let lo : Lattice n := fun _ => ⌊-R / r⌋ |
| let hi : Lattice n := fun _ => ⌊R / r⌋ |
| apply (Set.finite_Icc lo hi).subset |
| rintro z ⟨x, hx, rfl⟩ |
| have hcoord (i : Fin n) : |x i| ≤ R := by |
| calc |
| |x i| = ‖x i‖ := by simp only [Real.norm_eq_abs] |
| _ ≤ ‖x‖ := norm_le_pi_norm x i |
| _ ≤ R := hR x hx |
| constructor |
| · intro i |
| change ⌊-R / r⌋ ≤ ⌊x i / r⌋ |
| exact Int.floor_mono (div_le_div_of_nonneg_right (abs_le.mp (hcoord i)).1 hr.le) |
| · intro i |
| change ⌊x i / r⌋ ≤ ⌊R / r⌋ |
| exact Int.floor_mono (div_le_div_of_nonneg_right (abs_le.mp (hcoord i)).2 hr.le) |
|
|
| theorem centers_finite_on_bounded {n : ℕ} {C K : Set (Space n)} {r : ℝ} |
| (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖)) |
| (hK : Bornology.IsBounded K) : (C ∩ K).Finite := by |
| have hfin : (centerBoxIndex r '' (C ∩ K)).Finite := |
| (centerBoxIndex_image_finite hr hK).subset (Set.image_mono Set.inter_subset_right) |
| exact hfin.of_finite_image ((centerBoxIndex_injOn hr hsep).mono Set.inter_subset_left) |
|
|
| end PeriodicTilingThree |
|
|
| end |
|
|
| end OAI |
|
|
