File size: 4,810 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 | import OAI.Geometry.PeriodicTiling.Dependence
import Mathlib.GroupTheory.OrderOfElement
namespace OAI
noncomputable section
namespace PeriodicTilingThree
variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
def activeLabels (o : GraphOutputs E) (i : Channel p) (x : Plane) :
Finset (Label p i) := by
classical
exact Finset.univ.filter (Active E o i x)
@[simp] theorem mem_activeLabels (o : GraphOutputs E) (i : Channel p)
(x : Plane) (j : Label p i) :
j ∈ activeLabels E o i x ↔ Active E o i x j := by
classical
simp [activeLabels]
def selectSymbol (S : Finset (Symbol p)) : Symbol p := by
classical
exact if h : S.Nonempty then h.choose else 1
theorem selectSymbol_mem
{p : ℕ} [NeZero p]
(S : Finset (Symbol p)) (hS : S.Nonempty) :
selectSymbol S ∈ S := by
classical
simp only [selectSymbol, dite_eq_left hS]
exact hS.choose_spec
def selectedWord (o : GraphOutputs E) : WordArray p :=
fun n m => selectSymbol (activeLabels E o (.inl n) (0, m))
theorem selectedWord_active (o : GraphOutputs E)
(hactive : ∀ n x, ∃ j, Active E o (.inl n) x j) (n : Column p) (m : ℤ) :
Active E o (.inl n) (0, m) (selectedWord E o n m) := by
apply (mem_activeLabels E o (.inl n) (0, m) _).mp
apply selectSymbol_mem
obtain ⟨j, hj⟩ := hactive n (0, m)
exact ⟨j, (mem_activeLabels E o (.inl n) (0, m) j).mpr hj⟩
theorem selectedWord_active_at (o : GraphOutputs E) (hdep : HasDependence E o)
(hactive : ∀ n x, ∃ j, Active E o (.inl n) x j) (n : Column p) (x : Plane) :
Active E o (.inl n) x (selectedWord E o n (lineValue n x)) := by
apply (active_ordinary_iff E hdep n (y := (0, lineValue n x))
(by simp) _).mpr
exact selectedWord_active E o hactive n _
theorem selectedWord_lineRule (o : GraphOutputs E) (hdep : HasDependence E o)
(hactive : ∀ n x, ∃ j, Active E o (.inl n) x j)
(hwords : ∀ x (w : Word p), (∀ n, Active E o (.inl n) x (w n)) → Allowed p w) :
LineRule p (selectedWord E o) := by
intro d e
apply hwords (d, e)
intro n
simpa only [lineValue, mul_comm, add_comm] using
selectedWord_active_at E o hdep hactive n (d, e)
theorem selectedWord_nonconstant_of_forced_letters
(o : GraphOutputs E) (hdep : HasDependence E o)
(hactive : ∀ n x, ∃ j, Active E o (.inl n) x j)
(a b : Symbol p) (hab : a ≠ b) (x y : Plane)
(hx : ∀ n j, Active E o (.inl n) x j → j = a)
(hy : ∀ n j, Active E o (.inl n) y j → j = b) :
NonconstantColumns (selectedWord E o) := by
intro n
refine ⟨lineValue n x, lineValue n y, ?_⟩
have ha := hx n _ (selectedWord_active_at E o hdep hactive n x)
have hb := hy n _ (selectedWord_active_at E o hdep hactive n y)
intro he
exact hab (ha.symm.trans (he.trans hb))
theorem graph_period_useful (o : GraphOutputs E) (v : Plane)
(hv : Period (graph o) (v, (0 : V E))) (x : Plane) (k : Input E) :
o.c (x + v, k) = o.c (x, k) := by
have hm : point o (x, k) + (v, (0 : V E)) ∈ graph o :=
(hv (point o (x, k))).mpr (point_mem_graph o (x, k))
have he := eq_point_of_mem_graph o hm
have hq : q0 E (point o (x, k) + (v, (0 : V E))) = (x + v, k) := by
rw [map_add, q0_point]
refine Prod.ext rfl ?_
funext i
simp [q0]
rw [hq] at he
funext i
have hi := congrArg (fun z : Ambient E => (z.2.2 i).2) he
simpa only [Prod.fst_add, Prod.snd_add, Pi.add_apply, point_c,
Prod.fst_zero, Prod.snd_zero, Pi.zero_apply, add_zero] using hi.symm
theorem activeLabels_eq_of_graph_period (o : GraphOutputs E) (v : Plane)
(hv : Period (graph o) (v, (0 : V E))) (i : Channel p) (x : Plane) :
activeLabels E o i (x + v) = activeLabels E o i x := by
classical
have hc : usefulAt E o i (x + v) = usefulAt E o i x := by
funext w
exact congrFun (graph_period_useful E o v hv x (Function.update 0 i w)) i
ext j
simp only [mem_activeLabels, Active, hc]
theorem selectedWord_period_of_graph_period (o : GraphOutputs E) (M : ℤ)
(hv : Period (graph o) ((0, M), (0 : V E))) :
VerticalPeriod (selectedWord E o) M := by
intro n m
have hs := activeLabels_eq_of_graph_period E o (0, M) hv (.inl n) (0, m)
have hs' : activeLabels E o (.inl n) (0, m + M) =
activeLabels E o (.inl n) (0, m) := by simpa only [Prod.mk_add_mk, zero_add] using hs
exact congrArg selectSymbol hs'
theorem exists_positive_vertical_graph_period (o : GraphOutputs E)
(hper : FullyPeriodic (graph o)) :
∃ M : ℕ, 0 < M ∧ Period (graph o) ((0, (M : ℤ)), (0 : V E)) := by
obtain ⟨Λ, hΛ, hperiods⟩ := hper
refine ⟨Λ.index, Nat.pos_of_ne_zero hΛ.index_ne_zero, ?_⟩
have h := hperiods _ (Λ.nsmul_index_mem (((0, 1) : Plane), (0 : V E)))
simpa only [Prod.smul_mk, smul_zero, nsmul_eq_mul, mul_one] using h
end PeriodicTilingThree
end
end OAI
|