| import OAI.Geometry.PeriodicTiling.EncodedSystem | |
| import OAI.Geometry.PeriodicTiling.Activation | |
| import OAI.Geometry.PeriodicTiling.SystemObstruction | |
| namespace OAI | |
| namespace PeriodicTilingThree | |
| variable {p : ℕ} [NeZero p] (E : EncodingParameters p) | |
| theorem solves_not_fully_periodic {A : Set (Ambient E)} (hA : Solves E A) : | |
| ¬ FullyPeriodic A := by | |
| obtain ⟨o, rfl⟩ := exists_graph_of_solves E hA | |
| obtain ⟨_hg, hd, hw, hs, ho, ht⟩ := (solves_iff E (graph o)).mp hA | |
| have hdep : HasDependence E o := (dependenceTiles_iff E o).mp hd | |
| apply graph_not_fully_periodic_of_activity E o hdep | |
| · exact (wordConstraintTiles_iff E o hdep).mp hw | |
| · exact (seedConstraintTiles_iff E o hdep).mp hs | |
| · intro n | |
| exact ordinary_active_of_tile E o hdep n (ho n) | |
| · intro t | |
| exact seed_active_somewhere_of_tile E o hdep t (ht t) | |
| theorem testTile_not_fully_periodic {A : Set (Ambient E)} | |
| (hA : ∀ i : TestIndex E, Tiles (testTile E i) A) : ¬ FullyPeriodic A := | |
| solves_not_fully_periodic E ((solves_iff_index E A).mpr hA) | |
| theorem no_periodic_common_solution : | |
| ¬ ∃ A : Set (Ambient E), Solves E A ∧ FullyPeriodic A := by | |
| rintro ⟨A, hA, hperiod⟩ | |
| exact solves_not_fully_periodic E hA hperiod | |
| end PeriodicTilingThree | |
| end OAI | |