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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | 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
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