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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 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 | import OAI.Geometry.PeriodicTiling.ConstraintCycles
import OAI.Geometry.PeriodicTiling.ActivationTiles
namespace OAI
noncomputable section
namespace PeriodicTilingThree
variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
def testTiles : Finset (Finset (Ambient E)) := by
classical
exact {kernelTile E} ∪ dependenceTiles E ∪ wordConstraintTiles E ∪
seedConstraintTiles E ∪ Finset.univ.image (ordinaryActivationTile E) ∪
Finset.univ.image (seedActivationTile E)
theorem mem_testTiles_iff (F : Finset (Ambient E)) :
F ∈ testTiles E ↔ F = kernelTile E ∨ F ∈ dependenceTiles E ∨
F ∈ wordConstraintTiles E ∨ F ∈ seedConstraintTiles E ∨
(∃ n : Column p, ordinaryActivationTile E n = F) ∨
(∃ t : Fin 2, seedActivationTile E t = F) := by
classical
simp only [testTiles, Finset.mem_union, Finset.mem_singleton, Finset.mem_image,
Finset.mem_univ, true_and, or_assoc]
theorem kernelTile_mem_testTiles : kernelTile E ∈ testTiles E :=
(mem_testTiles_iff E _).mpr (Or.inl rfl)
theorem dependenceTiles_subset_testTiles : dependenceTiles E ⊆ testTiles E := by
intro F hF
exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inl hF))
theorem wordConstraintTiles_subset_testTiles : wordConstraintTiles E ⊆ testTiles E := by
intro F hF
exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inr (Or.inl hF)))
theorem seedConstraintTiles_subset_testTiles : seedConstraintTiles E ⊆ testTiles E := by
intro F hF
exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inr (Or.inr (Or.inl hF))))
theorem ordinaryActivationTile_mem_testTiles (n : Column p) :
ordinaryActivationTile E n ∈ testTiles E :=
(mem_testTiles_iff E _).mpr
(Or.inr (Or.inr (Or.inr (Or.inr (Or.inl ⟨n, rfl⟩)))))
theorem seedActivationTile_mem_testTiles (t : Fin 2) :
seedActivationTile E t ∈ testTiles E :=
(mem_testTiles_iff E _).mpr
(Or.inr (Or.inr (Or.inr (Or.inr (Or.inr ⟨t, rfl⟩)))))
theorem testTiles_nonempty : (testTiles E).Nonempty :=
⟨kernelTile E, kernelTile_mem_testTiles E⟩
theorem nonempty_of_mem_testTiles {F : Finset (Ambient E)}
(hF : F ∈ testTiles E) : F.Nonempty := by
classical
rcases (mem_testTiles_iff E F).mp hF with rfl | hF | hF | hF | ⟨n, rfl⟩ | ⟨t, rfl⟩
· exact kernelTile_nonempty E
· obtain ⟨d, _hd, rfl⟩ := Finset.mem_image.mp hF
exact invarianceTile_nonempty E (dependenceChannel d) (dependenceShift E d)
· exact nonempty_of_mem_wordConstraintTiles E hF
· exact nonempty_of_mem_seedConstraintTiles E hF
· exact ordinaryActivationTile_nonempty E n
· exact seedActivationTile_nonempty E t
def Solves (A : Set (Ambient E)) : Prop :=
∀ F ∈ testTiles E, Tiles F A
theorem solves_iff (A : Set (Ambient E)) : Solves E A ↔
Tiles (kernelTile E) A ∧
(∀ F ∈ dependenceTiles E, Tiles F A) ∧
(∀ F ∈ wordConstraintTiles E, Tiles F A) ∧
(∀ F ∈ seedConstraintTiles E, Tiles F A) ∧
(∀ n, Tiles (ordinaryActivationTile E n) A) ∧
(∀ t, Tiles (seedActivationTile E t) A) := by
constructor
· intro h
exact ⟨h _ (kernelTile_mem_testTiles E),
fun F hF => h F (dependenceTiles_subset_testTiles E hF),
fun F hF => h F (wordConstraintTiles_subset_testTiles E hF),
fun F hF => h F (seedConstraintTiles_subset_testTiles E hF),
fun n => h _ (ordinaryActivationTile_mem_testTiles E n),
fun t => h _ (seedActivationTile_mem_testTiles E t)⟩
· rintro ⟨hg, hd, hw, hs, ha, hb⟩ F hF
rcases (mem_testTiles_iff E F).mp hF with rfl | hF | hF | hF | ⟨n, rfl⟩ | ⟨t, rfl⟩
· exact hg
· exact hd F hF
· exact hw F hF
· exact hs F hF
· exact ha n
· exact hb t
abbrev TestIndex := ↥(testTiles E)
def testTile (i : TestIndex E) : Finset (Ambient E) := i.val
theorem testTile_nonempty (i : TestIndex E) : (testTile E i).Nonempty :=
nonempty_of_mem_testTiles E i.property
instance testIndex_nonempty : Nonempty (TestIndex E) :=
⟨⟨kernelTile E, kernelTile_mem_testTiles E⟩⟩
theorem solves_iff_index (A : Set (Ambient E)) :
Solves E A ↔ ∀ i : TestIndex E, Tiles (testTile E i) A := by
constructor
· intro h i
exact h i.val i.property
· intro h F hF
exact h ⟨F, hF⟩
theorem exists_graph_of_solves {A : Set (Ambient E)} (hA : Solves E A) :
∃ o : GraphOutputs E, graph o = A :=
(kernelTile_iff_graph E A).mp (hA _ (kernelTile_mem_testTiles E))
end PeriodicTilingThree
end
end OAI
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