File size: 5,892 Bytes
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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.Geometry.PeriodicTiling.Stacking
import Mathlib.GroupTheory.SpecificGroups.Cyclic
namespace OAI
universe uV uι
namespace PeriodicTilingThree
theorem not_dvd_card_of_lt (V : Type uV) [AddCommGroup V] [Finite V]
{q : ℕ} (hq : Nat.card V < q) : ¬ q ∣ Nat.card V := by
intro hdiv
exact (Nat.not_le_of_gt hq) (Nat.le_of_dvd Nat.card_pos hdiv)
theorem cyclicStackOrder_pos (V : Type uV) [AddCommGroup V] [Finite V]
(q : ℕ) (hq : q.Prime) : 0 < q * Nat.card V :=
Nat.mul_pos hq.pos Nat.card_pos
noncomputable def freshCyclicEquiv
(V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
(q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V) :
V × ZMod q ≃+ ZMod (q * Nat.card V) := by
let eV : V ≃+ ZMod (Nat.card V) :=
(zmodAddCyclicAddEquiv (G := V) inferInstance).symm
let hcop : q.Coprime (Nat.card V) :=
hq.coprime_iff_not_dvd.mpr havoid
exact
((eV.prodCongr (AddEquiv.refl (ZMod q))).trans
(AddEquiv.prodComm :
ZMod (Nat.card V) × ZMod q ≃+
ZMod q × ZMod (Nat.card V))).trans
(ZMod.chineseRemainder hcop).toAddEquiv.symm
noncomputable def cyclicStackEquiv
(V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
(q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V) :
((Plane × V) × ZMod q) ≃+ Plane × ZMod (q * Nat.card V) :=
(AddEquiv.prodAssoc :
((Plane × V) × ZMod q) ≃+ Plane × (V × ZMod q)).trans
((AddEquiv.refl Plane).prodCongr (freshCyclicEquiv V q hq havoid))
@[simp]
theorem cyclicStackEquiv_fst
(V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
(q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
(x : (Plane × V) × ZMod q) :
(cyclicStackEquiv V q hq havoid x).1 = x.1.1 := rfl
section Transport
variable (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
variable (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
noncomputable def cyclicStackComplement (A : Set (Plane × V)) :
Set (Plane × ZMod (q * Nat.card V)) :=
cyclicStackEquiv V q hq havoid '' (A ×ˢ ({0} : Set (ZMod q)))
noncomputable def cyclicStackProjection
(B : Set (Plane × ZMod (q * Nat.card V))) : Set (Plane × V) :=
Prod.fst '' (cyclicStackEquiv V q hq havoid ⁻¹' B)
theorem fullyPeriodic_cyclicStackComplement {A : Set (Plane × V)}
(hA : FullyPeriodic A) :
FullyPeriodic (cyclicStackComplement V q hq havoid A) := by
let : NeZero q := ⟨hq.ne_zero⟩
exact (fullyPeriodic_image_addEquiv_iff
(cyclicStackEquiv V q hq havoid)).mpr hA.prod_singleton_zero
theorem fullyPeriodic_cyclicStackProjection
{B : Set (Plane × ZMod (q * Nat.card V))} (hB : FullyPeriodic B) :
FullyPeriodic (cyclicStackProjection V q hq havoid B) := by
exact FullyPeriodic.image_of_surjective
(AddMonoidHom.fst (Plane × V) (ZMod q))
(fun x => ⟨(x, 0), rfl⟩)
(FullyPeriodic.preimage_addEquiv (cyclicStackEquiv V q hq havoid) hB)
end Transport
section FiniteFamily
variable {ι : Type uι} [Fintype ι]
variable (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
variable (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
noncomputable def cyclicStackTile
(F : ι → Finset (Plane × V)) (E : ι → Finset (ZMod q)) :
Finset (Plane × ZMod (q * Nat.card V)) := by
classical
letI : NeZero q := ⟨hq.ne_zero⟩
exact (stack F E).image (cyclicStackEquiv V q hq havoid)
theorem tiles_cyclicStackTile
{F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
(hpart : IsColorPartition E) {A : Set (Plane × V)}
(hA : ∀ ν, Tiles (F ν) A) :
Tiles (cyclicStackTile V q hq havoid F E)
(cyclicStackComplement V q hq havoid A) := by
classical
let : NeZero q := ⟨hq.ne_zero⟩
exact Tiles.image_addEquiv (cyclicStackEquiv V q hq havoid)
(tiles_stack_lift hpart hA)
theorem tiles_stack_of_tiles_cyclicStackTile
{F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
{B : Set (Plane × ZMod (q * Nat.card V))}
(hB : Tiles (cyclicStackTile V q hq havoid F E) B) :
Tiles (stack F E) (cyclicStackEquiv V q hq havoid ⁻¹' B) := by
classical
let : NeZero q := ⟨hq.ne_zero⟩
simpa only [cyclicStackTile, Finset.image_image, Function.comp_def,
AddEquiv.symm_apply_apply, Finset.image_id'] using
Tiles.preimage_addEquiv (cyclicStackEquiv V q hq havoid) hB
theorem tiles_of_tiles_cyclicStackTile
{F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
(hpart : IsColorPartition E) (hfull : HasFullDifferences E)
(hE : ∀ ν, (E ν).Nonempty)
{B : Set (Plane × ZMod (q * Nat.card V))}
(hB : Tiles (cyclicStackTile V q hq havoid F E) B) :
∀ ν, Tiles (F ν) (cyclicStackProjection V q hq havoid B) := by
classical
let : NeZero q := ⟨hq.ne_zero⟩
exact tiles_of_tiles_stack hpart hfull hE
(tiles_stack_of_tiles_cyclicStackTile V q hq havoid hB)
theorem cyclicStack_counterexample_of_common_system
{F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
(hpart : IsColorPartition E) (hfull : HasFullDifferences E)
(hE : ∀ ν, (E ν).Nonempty)
(hcommon : ∃ A, ∀ ν, Tiles (F ν) A)
(hnone : ∀ A, (∀ ν, Tiles (F ν) A) → ¬ FullyPeriodic A) :
(cyclicStackTile V q hq havoid F E).Nonempty ∧
(∃ A, Tiles (cyclicStackTile V q hq havoid F E)
(cyclicStackComplement V q hq havoid A)) ∧
∀ B, Tiles (cyclicStackTile V q hq havoid F E) B →
¬ FullyPeriodic B := by
obtain ⟨A, hA⟩ := hcommon
have ht := tiles_cyclicStackTile V q hq havoid hpart hA
refine ⟨ht.tile_nonempty, ⟨A, ht⟩, ?_⟩
intro B hB hperiod
exact hnone (cyclicStackProjection V q hq havoid B)
(tiles_of_tiles_cyclicStackTile V q hq havoid hpart hfull hE hB)
(fullyPeriodic_cyclicStackProjection V q hq havoid hperiod)
end FiniteFamily
end PeriodicTilingThree
end OAI
|