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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.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
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