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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 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import OAI.Geometry.PeriodicTiling.MarkedTile
import OAI.Geometry.PeriodicTiling.VoxelMeasure
import Mathlib.MeasureTheory.Group.Measure
namespace OAI
noncomputable section
namespace PeriodicTilingThree.MarkedTile
open Set MeasureTheory
@[simp] theorem castLattice_scale (m : ℕ) (u : Lattice 3) :
castLattice (scale m u) = scaledCast m u := by
ext i
simp [scale, scaledCast, castLattice]
def center (m : ℕ) (a : Space 3) (u : Lattice 3) : Space 3 :=
a + scaledCast m u
@[simp] theorem center_zero (m : ℕ) (a : Space 3) : center m a 0 = a := by
simp [center]
def centers (m : ℕ) (U : Finset (Lattice 3)) (A : Set (Space 3)) : Set (Space 3) :=
Set.range (fun q : A × ↥U => center m q.1 q.2)
theorem mem_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)}
{c : Space 3} : c ∈ centers m U A ↔ ∃ a ∈ A, ∃ u ∈ U, center m a u = c := by
constructor
· rintro ⟨⟨a, u⟩, h⟩
exact ⟨a, a.property, u, u.property, h⟩
· rintro ⟨a, ha, u, hu, h⟩
exact ⟨(⟨a, ha⟩, ⟨u, hu⟩), h⟩
theorem subset_centers {m : ℕ} {U : Finset (Lattice 3)} (h0 : 0 ∈ U)
(A : Set (Space 3)) : A ⊆ centers m U A := by
intro a ha
exact mem_centers.mpr ⟨a, ha, 0, h0, center_zero m a⟩
namespace ResidueSection
variable {m : ℕ} [NeZero m] (s : ResidueSection m)
def constituent (w u : Lattice 3) : Finset (Lattice 3) :=
if u = 0 then s.T1 w else s.T0
def part (w u : Lattice 3) : Finset (Lattice 3) :=
(s.constituent w u).image (fun t => scale m u + t)
theorem mem_part {w u z : Lattice 3} :
z ∈ s.part w u ↔ ∃ v, s.point w u v = z := by
classical
by_cases hu : u = 0
· subst u
simp [part, constituent, mem_T1]
· simp only [part, constituent, hu, ite_false, Finset.mem_image]
constructor
· rintro ⟨t, ht, he⟩
obtain ⟨v, rfl⟩ := s.mem_T0.mp ht
exact ⟨v, by simpa only [s.point_of_ne_zero w hu] using he⟩
· rintro ⟨v, hv⟩
exact ⟨s.t v, s.mem_T0.mpr ⟨v, rfl⟩,
by simpa only [s.point_of_ne_zero w hu] using hv⟩
theorem part_subset_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
{u : Lattice 3} (hu : u ∈ U) : s.part w u ⊆ s.finalTile w U := by
intro z hz
obtain ⟨v, hv⟩ := s.mem_part.mp hz
exact s.mem_finalTile.mpr ⟨u, hu, v, hv⟩
theorem parts_disjoint {w : Lattice 3} {U : Finset (Lattice 3)} (hw : w ∉ U)
{u u' : ↥U} (hne : u ≠ u') : Disjoint (s.part w u) (s.part w u') := by
classical
apply Finset.disjoint_left.mpr
intro z hz hz'
obtain ⟨v, hv⟩ := s.mem_part.mp hz
obtain ⟨v', hv'⟩ := s.mem_part.mp hz'
have hp : (u, v) = (u', v') := s.point_injective hw (hv.trans hv'.symm)
exact hne (congrArg Prod.fst hp)
theorem mem_thickening_part {w u : Lattice 3} {x : Space 3} :
x ∈ Thickening (s.part w u) ↔
x - scaledCast m u ∈ Thickening (s.constituent w u) := by
classical
constructor
· rintro ⟨z, hz, hx⟩
obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hz
refine ⟨t, ht, ?_⟩
simpa only [map_add, castLattice_scale, sub_add_eq_sub_sub] using hx
· rintro ⟨t, ht, hx⟩
refine ⟨scale m u + t, Finset.mem_image.mpr ⟨t, ht, rfl⟩, ?_⟩
simpa only [map_add, castLattice_scale, sub_add_eq_sub_sub] using hx
theorem mem_translated_part {w u : Lattice 3} {a x : Space 3} :
x - a ∈ Thickening (s.part w u) ↔
x - center m a u ∈ Thickening (s.constituent w u) := by
simpa only [center, sub_add_eq_sub_sub] using
(s.mem_thickening_part (w := w) (u := u) (x := x - a))
theorem mem_thickening_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
{x : Space 3} :
x ∈ Thickening (s.finalTile w U) ↔
∃ u : ↥U, x ∈ Thickening (s.part w u) := by
constructor
· rintro ⟨z, hz, hx⟩
obtain ⟨u, hu, v, hv⟩ := s.mem_finalTile.mp hz
exact ⟨⟨u, hu⟩, z, s.mem_part.mpr ⟨v, hv⟩, hx⟩
· rintro ⟨u, z, hz, hx⟩
exact ⟨z, s.part_subset_finalTile u.property hz, hx⟩
theorem mem_translated_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
{a x : Space 3} :
x - a ∈ Thickening (s.finalTile w U) ↔
∃ u : ↥U, x - center m a u ∈ Thickening (s.constituent w u) := by
rw [s.mem_thickening_finalTile]
exact exists_congr fun u => s.mem_translated_part
theorem Tstar_subset_constituent (w u : Lattice 3) :
s.Tstar ⊆ s.constituent w u := by
classical
intro z hz
by_cases hu : u = 0
· simp only [constituent, hu, ite_true, s.T1_eq_insert]
exact Finset.mem_insert_of_mem hz
· simpa only [constituent, hu, ite_false] using (Finset.mem_erase.mp hz).2
theorem Tstar_nonempty (hm : 2 ≤ m) : s.Tstar.Nonempty := by
apply Finset.card_pos.mp
rw [s.card_Tstar]
have hp := Nat.pow_le_pow_left hm 3
exact Nat.sub_pos_of_lt ((by decide : 1 < 2 ^ 3).trans_le hp)
theorem common_voxel_subset_constituent (w u : Lattice 3) (a : Space 3)
{t : Lattice 3} (ht : t ∈ s.Tstar) :
unitVoxel (center m a u + castLattice t) ⊆
{x | x - center m a u ∈ Thickening (s.constituent w u)} := by
intro x hx
refine ⟨t, s.Tstar_subset_constituent w u ht, ?_⟩
simpa only [mem_unitVoxel, sub_add_eq_sub_sub] using hx
end ResidueSection
end PeriodicTilingThree.MarkedTile
end
end OAI
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