| import OAI.Geometry.PeriodicTiling.CommonModel |
| import OAI.Geometry.PeriodicTiling.EncodedSystem |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PeriodicTilingThree.CommonModel |
|
|
| variable {p : ℕ} [NeZero p] (E : EncodingParameters p) |
|
|
| theorem ordinaryActivationMap_outputs (n : Column p) (b : Base E) : |
| ordinaryActivationMap E n (outputs E) b = |
| ordinaryForward E n b.1 (b.2 (.inl n)) := by |
| funext j |
| simp [ordinaryActivationMap, ordinarySource, activationSource, outputs, ordinaryForward] |
|
|
| theorem seedActivationMap_outputs (t : Fin 2) (b : Base E) : |
| seedActivationMap E t (outputs E) b = seedActualForward E t b.1 b.2 := by |
| rfl |
|
|
| theorem common_solution : Solves E (commonGraph E) := by |
| apply (solves_iff E (commonGraph E)).mpr |
| refine ⟨commonGraph_tiles_kernel E, ?_, ?_, ?_, ?_, ?_⟩ |
| · exact (dependenceTiles_iff E (outputs E)).mpr (outputs_hasDependence E) |
| · exact (wordConstraintTiles_iff E (outputs E) (outputs_hasDependence E)).mpr |
| (active_word_allowed E) |
| · apply (seedConstraintTiles_iff E (outputs E) (outputs_hasDependence E)).mpr |
| intro t n x j ht hj |
| exact active_seed_forces_symbol E t n x j ht hj |
| · intro n |
| apply (ordinaryActivationTile_iff E n (outputs E)).mpr |
| intro b |
| rw [ordinaryActivationMap_outputs] |
| exact ordinaryForward_bijective E n b.1 (b.2 (.inl n)) |
| · intro t |
| apply (seedActivationTile_iff E t (outputs E)).mpr |
| intro b |
| rw [seedActivationMap_outputs] |
| exact seedActualForward_bijective E t b.1 b.2 |
|
|
| theorem commonGraph_tiles_testTile (i : TestIndex E) : |
| Tiles (testTile E i) (commonGraph E) := |
| (solves_iff_index E (commonGraph E)).mp (common_solution E) i |
|
|
| end PeriodicTilingThree.CommonModel |
|
|
| namespace PeriodicTilingThree |
|
|
| theorem exists_common_solution {p : ℕ} [NeZero p] (E : EncodingParameters p) : |
| ∃ A : Set (Ambient E), Solves E A := |
| ⟨CommonModel.commonGraph E, CommonModel.common_solution E⟩ |
|
|
| end PeriodicTilingThree |
|
|
| end |
|
|
| end OAI |
|
|
