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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 | 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
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