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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 37 38 39 40 41 42 43 44 45 | 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
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