| import OAI.Geometry.PeriodicTiling.CyclicStack |
| import OAI.Geometry.PeriodicTiling.FullDifferencePartition |
|
|
| namespace OAI |
|
|
| universe uι uV |
|
|
| namespace PeriodicTilingThree |
|
|
| theorem exists_cyclic_stack_of_system |
| {ι : Type uι} [Fintype ι] [Nonempty ι] |
| (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V] |
| (F : ι → Finset (Plane × V)) |
| (hcommon : ∃ A, ∀ ν, Tiles (F ν) A) |
| (hnone : ∀ A, (∀ ν, Tiles (F ν) A) → ¬ FullyPeriodic A) : |
| ∃ Q : ℕ, 0 < Q ∧ |
| ∃ T : Finset (Plane × ZMod Q), T.Nonempty ∧ |
| (∃ B, Tiles T B) ∧ ∀ B, Tiles T B → ¬ FullyPeriodic B := by |
| classical |
| let s := Fintype.card ι |
| have hs : 0 < s := Fintype.card_pos |
| let e : ι ≃ Fin s := Fintype.equivFin ι |
| let F' : Fin s → Finset (Plane × V) := fun ν => F (e.symm ν) |
| have hcommon' : ∃ A, ∀ ν, Tiles (F' ν) A := by |
| obtain ⟨A, hA⟩ := hcommon |
| exact ⟨A, fun ν => hA (e.symm ν)⟩ |
| have hnone' : ∀ A, (∀ ν, Tiles (F' ν) A) → ¬ FullyPeriodic A := by |
| intro A hA |
| apply hnone A |
| intro ν |
| simpa only [F', Equiv.symm_apply_apply] using hA (e ν) |
| obtain ⟨q, hq, hcard, _hfresh, E, hE, hpart, hfull⟩ := |
| exists_fresh_prime_fullDifference_partition s hs ∅ (Nat.card V) |
| let havoid : ¬ q ∣ Nat.card V := not_dvd_card_of_lt V hcard |
| have hc := cyclicStack_counterexample_of_common_system V q hq havoid |
| (F := F') (E := E) hpart hfull hE hcommon' hnone' |
| refine ⟨q * Nat.card V, cyclicStackOrder_pos V q hq, |
| cyclicStackTile V q hq havoid F' E, hc.1, ?_, hc.2.2⟩ |
| obtain ⟨A, hA⟩ := hc.2.1 |
| exact ⟨cyclicStackComplement V q hq havoid A, hA⟩ |
|
|
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
