| import OAI.Geometry.PeriodicTiling.CommonSolution |
| import OAI.Geometry.PeriodicTiling.EncodedSystemObstruction |
| import OAI.Geometry.PeriodicTiling.EncodingPrimeSelection |
| import OAI.Geometry.PeriodicTiling.CyclicFactorCRT |
| import OAI.Geometry.PeriodicTiling.CyclicSystemStacking |
|
|
| namespace OAI |
|
|
| namespace PeriodicTilingThree |
|
|
| theorem cyclic_counterexample_of_parameters {p : ℕ} [NeZero p] |
| (E : EncodingParameters p) : |
| ∃ Q : ℕ, 0 < Q ∧ |
| ∃ F : Finset (Plane × ZMod Q), F.Nonempty ∧ |
| (∃ A, Tiles F A) ∧ ∀ A, Tiles F A → ¬ FullyPeriodic A := by |
| classical |
| let : IsAddCyclic (V E) := isAddCyclic_of_surjective |
| (finiteFactorEquiv E).symm (finiteFactorEquiv E).symm.surjective |
| apply exists_cyclic_stack_of_system (V E) (testTile E) |
| · exact ⟨CommonModel.commonGraph E, CommonModel.commonGraph_tiles_testTile E⟩ |
| · intro A hA |
| exact testTile_not_fully_periodic E hA |
|
|
| theorem exists_cyclic_counterexample : |
| ∃ Q : ℕ, 0 < Q ∧ |
| ∃ F : Finset (Plane × ZMod Q), F.Nonempty ∧ |
| (∃ A, Tiles F A) ∧ ∀ A, Tiles F A → ¬ FullyPeriodic A := by |
| obtain ⟨p, hpbound, hp⟩ := Nat.exists_infinite_primes 201 |
| let : NeZero p := ⟨hp.ne_zero⟩ |
| let E : EncodingParameters p := encodingParameters hp (by omega) |
| exact cyclic_counterexample_of_parameters E |
|
|
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
