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92416a2 | 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 | 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
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