| import Mathlib |
| import OAI.Computability.VertexCover.PCP.PoweringTest |
| import OAI.Computability.VertexCover.PCP.AlphabetGraph |
| import OAI.Computability.VertexCover.PCP.ClauseVerifier |
| import OAI.Computability.VertexCover.PCP.FinalConstants |
|
|
| namespace OAI |
|
|
| |
|
|
| noncomputable section |
|
|
| namespace UniqueGames.Foundations.PCP.AmplificationRound |
|
|
| abbrev Label := QueryIncidence.Label 6 |
| abbrev Addresses := PoweringLabels.PortWords Preprocessing.Port FinalConstants.walkLength |
| abbrev PoweredAlphabet := |
| PoweringLabels.PaddedLabel Preprocessing.Port FinalConstants.walkLength Label |
| |
| instance : Fintype PoweredAlphabet := Fintype.ofFinite _ |
| instance : DecidableEq PoweredAlphabet := Classical.decEq _ |
| |
| variable {V E : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
| [Nonempty E] |
| |
| abbrev PoweredDart (G : ConstraintGraph V E Label) := |
| PoweringTest.Dart (Preprocessing.Vertex G) Preprocessing.Port |
| (2 * FinalConstants.endpointLength) |
| |
| instance (G : ConstraintGraph V E Label) : DecidableEq (PoweredDart G) := Classical.decEq _ |
| |
| def selectors (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (G : ConstraintGraph V E Label) (v : Preprocessing.Vertex G) : |
| PoweringLabels.AddressSelector (Preprocessing.portGraph G) |
| FinalConstants.walkLength v := |
| PoweringLabels.listSelector (Preprocessing.portGraph G) FinalConstants.walkLength |
| v addresses complete |
| |
| def powered (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (G : ConstraintGraph V E Label) : |
| ConstraintGraph (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet := |
| PoweringTest.poweredGraph (Preprocessing.portGraph G) (Preprocessing.graph G).accepts |
| (2 * FinalConstants.endpointLength) (selectors addresses complete G) |
| |
| abbrev Vertex (G : ConstraintGraph V E Label) := |
| QueryIncidence.Vertex |
| (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet) |
| (AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet) |
| |
| abbrev Dart (G : ConstraintGraph V E Label) := |
| QueryIncidence.Dart (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet) 6 |
| |
| instance (G : ConstraintGraph V E Label) : Fintype (Vertex G) := by |
| letI : Fintype (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet) := inferInstance |
| letI : Fintype |
| (AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet) := |
| inferInstance |
| change Fintype |
| (AlphabetGraph.Event (PoweredDart G) PoweredAlphabet ⊕ |
| AlphabetGraph.Address (Preprocessing.Vertex G) (PoweredDart G) PoweredAlphabet) |
| infer_instance |
| instance (G : ConstraintGraph V E Label) : Fintype (Dart G) := by |
| change Fintype ((AlphabetGraph.Event (PoweredDart G) PoweredAlphabet × Fin 6) × Bool) |
| infer_instance |
| instance (G : ConstraintGraph V E Label) : DecidableEq (Vertex G) := Classical.decEq _ |
| |
| def graph (addresses : List Addresses) (complete : ∀ w, w ∈ addresses) |
| (G : ConstraintGraph V E Label) : ConstraintGraph (Vertex G) (Dart G) Label := |
| AlphabetGraph.graph (powered addresses complete G) |
| |
| omit [Nonempty E] in |
| theorem powered_completeness (addresses : List Addresses) |
| (complete : ∀ w, w ∈ addresses) (G : ConstraintGraph V E Label) |
| (satisfied : G.Satisfiable) : (powered addresses complete G).Satisfiable := by |
| obtain ⟨assignment, hassignment⟩ := Preprocessing.completeness G satisfied |
| refine ⟨PoweringOpinions.honestLabels (Preprocessing.portGraph G) |
| FinalConstants.walkLength assignment, ?_⟩ |
| apply PoweringTest.perfect_completeness (Preprocessing.portGraph G) |
| (Preprocessing.graph G).accepts (2 * FinalConstants.endpointLength) |
| (selectors addresses complete G) assignment |
| intro e |
| exact hassignment e |
| |
| omit [Nonempty E] in |
| theorem completeness (addresses : List Addresses) |
| (complete : ∀ w, w ∈ addresses) (G : ConstraintGraph V E Label) |
| (satisfied : G.Satisfiable) : (graph addresses complete G).Satisfiable := |
| AlphabetGraph.perfect_completeness _ |
| (powered_completeness addresses complete G satisfied) |
| |
| end UniqueGames.Foundations.PCP.AmplificationRound |
| end |
| |
| end OAI |
| |
