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import OAI.Computability.VertexCover.PCP.GenericGraphTables
import OAI.Computability.VertexCover.PCP.AlphabetTableEnumeration
import OAI.Computability.VertexCover.PCP.AlphabetTableRelations
namespace OAI
namespace UniqueGames.Foundations.PCP.AlphabetTable.Table
variable {q : Nat}
def rowFromDart (input : GenericGraphTables.Table q)
(dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
GraphTables.DartRow (Enumeration.vertexCount input.vertices input.darts q)
(Enumeration.dartCount input.darts q) :=
let edge := dart.1.1.1
let event := dart.1.1.2
let slot := dart.1.2
let old := input.rows[edge]
let head := input.rows[old.reverseIndex].tail
{ tail := Enumeration.vertexEquiv input.vertices input.darts q
(if dart.2 then
.inr (Queries.globalize old.tail head edge
(Queries.query (GenericGraphTables.relationAt old.relation) event slot))
else .inl (edge, event))
reverseIndex := Enumeration.dartEquiv input.darts q (dart.1, !dart.2)
relation := Relations.relationFromTable old.relation (decide (old.tail = head))
event slot dart.2 }
def rows (input : GenericGraphTables.Table q) :
GraphTables.Rows (Enumeration.vertexCount input.vertices input.darts q)
(Enumeration.dartCount input.darts q) :=
Vector.ofFn (fun index =>
rowFromDart input ((Enumeration.dartEquiv input.darts q).symm index))
@[simp] theorem rows_at (input : GenericGraphTables.Table q)
(index : Fin (Enumeration.dartCount input.darts q)) :
(rows input)[index] =
rowFromDart input ((Enumeration.dartEquiv input.darts q).symm index) := by
simp [rows]
theorem rows_reverseIndex (input : GenericGraphTables.Table q)
(index : Fin (Enumeration.dartCount input.darts q)) :
(rows input)[index].reverseIndex =
Enumeration.dartEquiv input.darts q
(((Enumeration.dartEquiv input.darts q).symm index).1,
!((Enumeration.dartEquiv input.darts q).symm index).2) := by
simp [rows, rowFromDart]
private theorem dartRow_ext {n m : Nat} {left right : GraphTables.DartRow n m}
(ht : left.tail = right.tail) (hr : left.reverseIndex = right.reverseIndex)
(hp : left.relation = right.relation) : left = right := by
cases left
cases right
cases ht
cases hr
cases hp
rfl
private theorem relation_ext {left right : GraphTables.RelationTable}
(h : ∀ a b, GraphTables.relationAt left a b = GraphTables.relationAt right a b) :
left = right := by
apply Vector.ext
intro i hi
let labels := GraphTables.relationIndex.symm (⟨i, hi⟩ : Fin 4096)
have index : GraphTables.relationIndex labels = (⟨i, hi⟩ : Fin 4096) :=
GraphTables.relationIndex.apply_symm_apply _
have point := h labels.1 labels.2
simpa only [GraphTables.relationAt, index, Fin.getElem_fin] using point
theorem rowFromDart_eq_graphRow (input : GenericGraphTables.Table q)
(dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
rowFromDart input dart =
{ tail := Enumeration.vertexEquiv input.vertices input.darts q
((AlphabetGraph.graph (GenericGraphTables.semantics input)).tail dart)
reverseIndex := Enumeration.dartEquiv input.darts q
((AlphabetGraph.graph (GenericGraphTables.semantics input)).reverse dart)
relation := GraphTables.relationOf (fun a b =>
(AlphabetGraph.graph (GenericGraphTables.semantics input)).accepts dart
(Enumeration.labelEquiv.symm a) (Enumeration.labelEquiv.symm b)) } := by
rcases dart with ⟨⟨⟨edge, event⟩, slot⟩, orientation⟩
apply dartRow_ext
· cases orientation with
| false => rfl
| true =>
have query := Queries.globalize_query (GenericGraphTables.semantics input)
edge event slot
have predicate : (GenericGraphTables.semantics input).accepts edge =
GenericGraphTables.relationAt input.rows[edge].relation := rfl
rw [predicate] at query
simpa only [rowFromDart, AlphabetGraph.graph, QueryIncidence.graph,
QueryIncidence.tail, GenericGraphTables.semantics_tail,
ConstraintGraph.head, GenericGraphTables.semantics_reverse,
ite_true] using
congrArg (fun address => Enumeration.vertexEquiv input.vertices input.darts q
(.inr address)) query
· rfl
· apply relation_ext
intro a b
rw [GraphTables.relationAt_relationOf]
have relation := Relations.relationAt_relation_graph (GenericGraphTables.semantics input)
edge event slot orientation a b
have predicate : (GenericGraphTables.semantics input).accepts edge =
Relations.oldPredicate input.rows[edge].relation := rfl
rw [predicate] at relation
simp only [rowFromDart, Relations.relationFromTable,
GenericGraphTables.semantics_tail, ConstraintGraph.head,
GenericGraphTables.semantics_reverse] at relation ⊢
exact relation
theorem rows_eq_graphRows (input : GenericGraphTables.Table q) :
rows input = GraphTables.graphRows
(GraphTables.enumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
(Enumeration.vertexEquiv input.vertices input.darts q)
(Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv) := by
apply Vector.ext
intro i hi
simp only [rows, GraphTables.graphRows, Vector.getElem_ofFn]
exact rowFromDart_eq_graphRow input
((Enumeration.dartEquiv input.darts q).symm ⟨i, hi⟩)
theorem rows_valid (input : GenericGraphTables.Table q) : GraphTables.Valid (rows input) := by
rw [rows_eq_graphRows]
exact GraphTables.graphRows_valid _
def build (input : GenericGraphTables.Table q) : GraphTables.Table where
vertices := Enumeration.vertexCount input.vertices input.darts q
darts := Enumeration.dartCount input.darts q
rows := rows input
valid := rows_valid input
@[simp] theorem build_vertices (input : GenericGraphTables.Table q) :
(build input).vertices = Enumeration.vertexCount input.vertices input.darts q := rfl
@[simp] theorem build_darts (input : GenericGraphTables.Table q) :
(build input).darts = Enumeration.dartCount input.darts q := rfl
private theorem table_rows_congr {n m : Nat} {left right : GraphTables.Rows n m}
(h : left = right) (hl : GraphTables.Valid left) (hr : GraphTables.Valid right) :
(⟨n, m, left, hl⟩ : GraphTables.Table) = ⟨n, m, right, hr⟩ := by
subst right
rfl
private theorem semantics_rows_congr {n m : Nat} {left right : GraphTables.Rows n m}
(h : left = right) (hl : GraphTables.Valid left) (hr : GraphTables.Valid right) :
GraphTables.semantics (⟨n, m, left, hl⟩ : GraphTables.Table) =
GraphTables.semantics (⟨n, m, right, hr⟩ : GraphTables.Table) := by
subst right
rfl
theorem build_eq_ofEnumeratedGraph (input : GenericGraphTables.Table q) :
build input =
GraphTables.ofEnumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
(Enumeration.vertexEquiv input.vertices input.darts q)
(Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv := by
exact table_rows_congr (rows_eq_graphRows input) (rows_valid input)
(GraphTables.graphRows_valid _)
theorem semantics_build (input : GenericGraphTables.Table q) :
GraphTables.semantics (build input) =
GraphTables.enumeratedGraph (AlphabetGraph.graph (GenericGraphTables.semantics input))
(Enumeration.vertexEquiv input.vertices input.darts q)
(Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv := by
exact (semantics_rows_congr (rows_eq_graphRows input) (rows_valid input)
(GraphTables.graphRows_valid _)).trans (GraphTables.semantics_ofGraph _)
theorem decode_build (input : GenericGraphTables.Table q) :
GraphTables.decodeTableBits (GraphTables.tableBits (build input)) =
some (GraphTables.ofEnumeratedGraph
(AlphabetGraph.graph (GenericGraphTables.semantics input))
(Enumeration.vertexEquiv input.vertices input.darts q)
(Enumeration.dartEquiv input.darts q) Enumeration.labelEquiv) := by
rw [GraphTables.decodeTableBits_encoded, build_eq_ofEnumeratedGraph]
theorem edgeSatisfied_build (input : GenericGraphTables.Table q)
(labeling : QueryIncidence.Vertex (AlphabetGraph.Event (Fin input.darts) (Fin q))
(AlphabetGraph.Address (Fin input.vertices) (Fin input.darts) (Fin q)) →
QueryIncidence.Label 6)
(dart : QueryIncidence.Dart (AlphabetGraph.Event (Fin input.darts) (Fin q)) 6) :
(GraphTables.semantics (build input)).edgeSatisfied
(fun v => Enumeration.labelEquiv
(labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v)))
(Enumeration.dartEquiv input.darts q dart) =
(AlphabetGraph.graph (GenericGraphTables.semantics input)).edgeSatisfied labeling dart := by
rw [semantics_build]
exact GraphTables.enumeratedGraph_edgeSatisfied _ _ _ _ labeling dart
theorem rejectionCount_build (input : GenericGraphTables.Table q)
(labeling : QueryIncidence.Vertex (AlphabetGraph.Event (Fin input.darts) (Fin q))
(AlphabetGraph.Address (Fin input.vertices) (Fin input.darts) (Fin q)) →
QueryIncidence.Label 6) :
(GraphTables.semantics (build input)).rejectionCount
(fun v => Enumeration.labelEquiv
(labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) =
(AlphabetGraph.graph (GenericGraphTables.semantics input)).rejectionCount labeling := by
rw [semantics_build]
exact GraphTables.enumeratedGraph_rejectionCount _ _ _ _ labeling
theorem perfect_completeness [Nonempty (Fin q)] (input : GenericGraphTables.Table q)
(satisfied : (GenericGraphTables.semantics input).Satisfiable) :
(GraphTables.semantics (build input)).Satisfiable := by
obtain ⟨labeling, accepted⟩ :=
AlphabetGraph.perfect_completeness (GenericGraphTables.semantics input) satisfied
refine ⟨fun v => Enumeration.labelEquiv
(labeling ((Enumeration.vertexEquiv input.vertices input.darts q).symm v)), ?_⟩
intro index
have h := edgeSatisfied_build input labeling
((Enumeration.dartEquiv input.darts q).symm index)
have index_eq := (Enumeration.dartEquiv input.darts q).apply_symm_apply index
rw [index_eq] at h
exact h.trans (accepted _)
theorem gap_transfer [Nonempty (Fin q)] (input : GenericGraphTables.Table q) (a b : Nat)
(source : ∀ labeling : Fin input.vertices → Fin q,
a * input.darts ≤ b * (GenericGraphTables.semantics input).rejectionCount labeling)
(labeling : Fin (build input).vertices → GraphTables.Label) :
a * (build input).darts ≤
(b * 12288) * (GraphTables.semantics (build input)).rejectionCount labeling := by
classical
let lifted := fun vertex => Enumeration.labelEquiv.symm
(labeling (Enumeration.vertexEquiv input.vertices input.darts q vertex))
have h := AlphabetGraph.gap_transfer (GenericGraphTables.semantics input) a b
(by simpa only [Fintype.card_fin] using source) lifted
rw [Enumeration.card_dart] at h
have counts := rejectionCount_build input lifted
have restored : (fun v : Fin (build input).vertices => Enumeration.labelEquiv
(lifted ((Enumeration.vertexEquiv input.vertices input.darts q).symm v))) = labeling := by
funext v
dsimp only [lifted]
rw [Enumeration.labelEquiv.apply_symm_apply]
exact congrArg labeling
((Enumeration.vertexEquiv input.vertices input.darts q).apply_symm_apply v)
rw [restored] at counts
rw [← counts] at h
exact h
theorem rowList_eq_ordered (input : GenericGraphTables.Table q) :
GraphTables.rowList (build input) =
(Enumeration.ordered (Enumeration.dartEquiv input.darts q)).map
(rowFromDart input) := by
change (Vector.ofFn (fun index => rowFromDart input
((Enumeration.dartEquiv input.darts q).symm index))).toList =
(List.ofFn (Enumeration.dartEquiv input.darts q).symm).map (rowFromDart input)
simp only [Vector.toList_ofFn, List.map_ofFn, Function.comp_def]
theorem tableWords_eq_ordered (input : GenericGraphTables.Table q) :
GraphTables.tableWords (build input) =
[Enumeration.vertexCount input.vertices input.darts q,
Enumeration.dartCount input.darts q] ++
(Enumeration.ordered (Enumeration.dartEquiv input.darts q)).flatMap
(fun dart => GraphTables.rowWords (rowFromDart input dart)) := by
simp only [GraphTables.tableWords, rowList_eq_ordered, build_vertices, build_darts,
List.flatMap_map]
@[simp] theorem rowList_length (input : GenericGraphTables.Table q) :
(GraphTables.rowList (build input)).length = Enumeration.dartCount input.darts q := by
exact GraphTables.rowList_length (build input)
theorem tableWords_length (input : GenericGraphTables.Table q) :
(GraphTables.tableWords (build input)).length =
2 + 4098 * Enumeration.dartCount input.darts q := by
exact GraphTables.tableWords_length (build input)
theorem tableBits_length_le (input : GenericGraphTables.Table q) :
(GraphTables.tableBits (build input)).length ≤
Enumeration.vertexCount input.vertices input.darts q +
Enumeration.dartCount input.darts q + 2 + Enumeration.dartCount input.darts q *
(Enumeration.vertexCount input.vertices input.darts q +
Enumeration.dartCount input.darts q + 8192) := by
exact GraphTables.tableBits_length_le (build input)
end UniqueGames.Foundations.PCP.AlphabetTable.Table
end OAI
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