| 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 | |