File size: 3,030 Bytes
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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | 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
|