| import OAI.Geometry.PeriodicTiling.ConstraintCycles |
| import OAI.Geometry.PeriodicTiling.ActivationTiles |
|
|
| namespace OAI |
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| noncomputable section |
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| namespace PeriodicTilingThree |
|
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| variable {p : ℕ} [NeZero p] (E : EncodingParameters p) |
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| 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) |
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| 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) |
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| 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))) |
|
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| 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⟩))))) |
|
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| 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⟩ |
|
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| 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 |
|
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| 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) |
|
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| def testTile (i : TestIndex E) : Finset (Ambient E) := i.val |
|
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| theorem testTile_nonempty (i : TestIndex E) : (testTile E i).Nonempty := |
| nonempty_of_mem_testTiles E i.property |
|
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| instance testIndex_nonempty : Nonempty (TestIndex E) := |
| ⟨⟨kernelTile E, kernelTile_mem_testTiles E⟩⟩ |
|
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| 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 |
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| end |
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| end OAI |
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|
