| import OAI.Geometry.PeriodicTiling.ConstituentGeometry |
| import OAI.Geometry.PeriodicTiling.AETilingCountable |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree.MarkedTile |
|
|
| open Set MeasureTheory Filter |
|
|
| namespace ResidueSection |
|
|
| variable {m : ℕ} [NeZero m] (s : ResidueSection m) |
|
|
| theorem translated_parts_unique_ae {w : Lattice 3} {U : Finset (Lattice 3)} |
| (hw : w ∉ U) (a : Space 3) (u u' : ↥U) : |
| ∀ᵐ x ∂volume, |
| x - center m a u ∈ Thickening (s.constituent w u) → |
| x - center m a u' ∈ Thickening (s.constituent w u') → u = u' := by |
| classical |
| by_cases he : u = u' |
| · exact Eventually.of_forall (fun _ _ _ => he) |
| · have hnull := thickening_inter_measure_zero_of_disjoint (s.parts_disjoint hw he) |
| have havoid : ∀ᵐ x ∂volume, |
| x ∉ Thickening (s.part w u) ∩ Thickening (s.part w u') := |
| compl_mem_ae_iff.mpr hnull |
| have hshift := (measurePreserving_sub_right volume a).quasiMeasurePreserving.ae havoid |
| filter_upwards [hshift] with x hx |
| intro hu hu' |
| exact False.elim (hx ⟨s.mem_translated_part.mpr hu, |
| s.mem_translated_part.mpr hu'⟩) |
|
|
| theorem indexed_constituents_ae {w : Lattice 3} {U : Finset (Lattice 3)} |
| (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)} |
| (htile : AETiles (Thickening (s.finalTile w U)) A) : |
| ∀ᵐ x ∂volume, ∃! q : A × ↥U, |
| x - center m q.1 q.2 ∈ Thickening (s.constituent w q.2) := by |
| classical |
| have hcount := htile.countable_of_thickening (s.finalTile_nonempty w h0) |
| let : Countable A := hcount.to_subtype |
| have hall : ∀ᵐ x ∂volume, ∀ (a : A) (u u' : ↥U), |
| x - center m a u ∈ Thickening (s.constituent w u) → |
| x - center m a u' ∈ Thickening (s.constituent w u') → u = u' := by |
| apply ae_all_iff.mpr |
| intro a |
| apply ae_all_iff.mpr |
| intro u |
| apply ae_all_iff.mpr |
| intro u' |
| exact s.translated_parts_unique_ae hw a u u' |
| filter_upwards [htile, hall] with x hx hpieces |
| obtain ⟨a, ha, haunique⟩ := hx |
| obtain ⟨u, hu⟩ := s.mem_translated_finalTile.mp ha |
| refine ⟨(a, u), hu, ?_⟩ |
| rintro ⟨a', u'⟩ hq |
| have ha' : a' = a := haunique a' (s.mem_translated_finalTile.mpr ⟨u', hq⟩) |
| subst a' |
| exact Prod.ext rfl (hpieces a u' u hq hu) |
|
|
| theorem center_injective {w : Lattice 3} {U : Finset (Lattice 3)} |
| (hm : 2 ≤ m) (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)} |
| (htile : AETiles (Thickening (s.finalTile w U)) A) : |
| Function.Injective (fun q : A × ↥U => center m q.1 q.2) := by |
| classical |
| have hcover := s.indexed_constituents_ae h0 hw htile |
| obtain ⟨t, ht⟩ := s.Tstar_nonempty hm |
| intro q q' he |
| by_contra hne |
| have hnull : volume (unitVoxel (center m q.1 q.2 + castLattice t)) = 0 := by |
| apply measure_mono_null ?_ (ae_iff.mp hcover) |
| intro x hx huniq |
| obtain ⟨r, _hr, hruniq⟩ := huniq |
| have hq : x - center m q.1 q.2 ∈ Thickening (s.constituent w q.2) := |
| s.common_voxel_subset_constituent w q.2 q.1 ht hx |
| have hq' : x - center m q'.1 q'.2 ∈ Thickening (s.constituent w q'.2) := by |
| apply s.common_voxel_subset_constituent w q'.2 q'.1 ht |
| simpa only [he] using hx |
| exact hne ((hruniq q hq).trans (hruniq q' hq').symm) |
| rw [unitVoxel_volume] at hnull |
| exact one_ne_zero hnull |
| |
| theorem center_mem_A_iff |
| {m : ℕ} [NeZero m] |
| {U : Finset (Lattice 3)} (h0 : 0 ∈ U) |
| {A : Set (Space 3)} |
| (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2)) |
| (q : A × ↥U) : center m q.1 q.2 ∈ A ↔ (q.2 : Lattice 3) = 0 := by |
| constructor |
| · intro hc |
| have he : q = ((⟨center m q.1 q.2, hc⟩ : A), (⟨0, h0⟩ : ↥U)) := by |
| apply hinj |
| simp |
| exact congrArg (fun p : A × ↥U => (p.2 : Lattice 3)) he |
| · intro hu |
| simpa only [hu, center_zero] using q.1.property |
| |
| open scoped Classical in |
| theorem constituent_type_eq {w : Lattice 3} {U : Finset (Lattice 3)} (h0 : 0 ∈ U) |
| {A : Set (Space 3)} |
| (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2)) |
| (q : A × ↥U) : |
| (if center m q.1 q.2 ∈ A then Thickening (s.T1 w) else Thickening s.T0) = |
| Thickening (s.constituent w q.2) := by |
| classical |
| by_cases hu : (q.2 : Lattice 3) = 0 |
| · rw [ite_eq_left ((center_mem_A_iff h0 hinj q).mpr hu)] |
| simp only [constituent, ite_eq_left hu] |
| · have hc : center m q.1 q.2 ∉ A := fun h => hu ((center_mem_A_iff h0 hinj q).mp h) |
| simp only [hc, ite_false, constituent, hu] |
| |
| theorem typed_constituents {w : Lattice 3} {U : Finset (Lattice 3)} |
| (hm : 2 ≤ m) (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)} |
| (htile : AETiles (Thickening (s.finalTile w U)) A) : |
| TypedAETiles (Thickening s.T0) (Thickening (s.T1 w)) (centers m U A) A := by |
| classical |
| have hinj := s.center_injective hm h0 hw htile |
| have hcover := s.indexed_constituents_ae h0 hw htile |
| filter_upwards [hcover] with x hx |
| obtain ⟨q, hq, hqunique⟩ := hx |
| let c : centers m U A := ⟨center m q.1 q.2, ⟨q, rfl⟩⟩ |
| refine ⟨c, ?_, ?_⟩ |
| · change x - center m q.1 q.2 ∈ |
| if center m q.1 q.2 ∈ A then Thickening (s.T1 w) else Thickening s.T0 |
| rw [s.constituent_type_eq h0 hinj q] |
| exact hq |
| · intro c' hc' |
| obtain ⟨q', he⟩ := c'.property |
| change center m q'.1 q'.2 = (c' : Space 3) at he |
| have hq' : x - center m q'.1 q'.2 ∈ Thickening (s.constituent w q'.2) := by |
| rw [← s.constituent_type_eq h0 hinj q', he] |
| exact hc' |
| have hqq := hqunique q' hq' |
| apply Subtype.ext |
| change (c' : Space 3) = center m q.1 q.2 |
| rw [← he, hqq] |
| |
| end ResidueSection |
| |
| theorem period_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)} |
| {v : Space 3} (hA : Period A v) : Period (centers m U A) v := by |
| intro c |
| constructor |
| · intro hc |
| obtain ⟨a, ha, u, hu, he⟩ := mem_centers.mp hc |
| have hav : a - v ∈ A := (hA (a - v)).mp (by simpa using ha) |
| refine mem_centers.mpr ⟨a - v, hav, u, hu, ?_⟩ |
| calc |
| center m (a - v) u = center m a u - v := by unfold center; abel |
| _ = c := by rw [he]; abel |
| · intro hc |
| obtain ⟨a, ha, u, hu, he⟩ := mem_centers.mp hc |
| refine mem_centers.mpr ⟨a + v, (hA a).mpr ha, u, hu, ?_⟩ |
| calc |
| center m (a + v) u = center m a u + v := by unfold center; abel |
| _ = c + v := by rw [he] |
| |
| theorem fullyPeriodic_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)} |
| (hA : EuclideanFullyPeriodic A) : EuclideanFullyPeriodic (centers m U A) := by |
| obtain ⟨b, hb⟩ := hA |
| exact ⟨b, fun v hv => period_centers (hb v hv)⟩ |
| |
| theorem tiles_of_centers_grid {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)} |
| {c0 : Space 3} |
| (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2)) |
| (hgrid : centers m U A = Set.range (fun z : Lattice 3 => c0 + scaledCast m z)) : |
| Tiles U {z : Lattice 3 | c0 + scaledCast m z ∈ A} := by |
| classical |
| rw [tiles_iff_unique_tile] |
| intro z |
| have hz : c0 + scaledCast m z ∈ centers m U A := by |
| rw [hgrid] |
| exact ⟨z, rfl⟩ |
| obtain ⟨q, hq⟩ := hz |
| have hd : c0 + scaledCast m (z - (q.2 : Lattice 3)) = (q.1 : Space 3) := by |
| have he : (q.1 : Space 3) + scaledCast m (q.2 : Lattice 3) = |
| c0 + scaledCast m z := hq |
| rw [map_sub] |
| calc |
| c0 + (scaledCast m z - scaledCast m (q.2 : Lattice 3)) = |
| (c0 + scaledCast m z) - scaledCast m (q.2 : Lattice 3) := by abel |
| _ = (q.1 : Space 3) := by rw [← he]; abel |
| refine ⟨q.2, ?_, ?_⟩ |
| · change c0 + scaledCast m (z - (q.2 : Lattice 3)) ∈ A |
| rw [hd] |
| exact q.1.property |
| · intro u hu |
| let a : A := ⟨c0 + scaledCast m (z - (u : Lattice 3)), hu⟩ |
| have hc : center m a u = c0 + scaledCast m z := by |
| change (c0 + scaledCast m (z - (u : Lattice 3))) + scaledCast m u = _ |
| rw [map_sub] |
| abel |
| have he : (a, u) = q := hinj (hc.trans hq.symm) |
| exact congrArg Prod.snd he |
| |
| end PeriodicTilingThree.MarkedTile |
| |
| end |
| |
| end OAI |
| |
