| import Mathlib |
| import OAI.Computability.VertexCover.Machines.Codec |
| import OAI.Computability.VertexCover.Machines.AlphabetFrame |
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| namespace OAI |
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| namespace VertexCover.Machine.AlphabetMachine |
| open UniqueGames.Foundations.PCP |
| open AlphabetTable Enumeration |
|
|
| def queryRank {q : ℕ} (s : State q) : Queries.RawQuery q → ℕ |
| | .inl (side,tape) => (if side then head s else (old s).1.1)*tapeCount q+(cubeEquiv q tape).val |
| | .inr tape => s.1.vertices*tapeCount q + (s.2*pairTapeCount q+(pairCubeEquiv q tape).val) |
| noncomputable def queryRankPoly {q : ℕ} (r : Queries.RawQuery q) : |
| Poly (code (q := q)) natBits (fun s => queryRank s r) := by |
| cases r with |
| | inl p => |
| let endpoint : Poly (code (q := q)) natBits (fun s => if p.1 then head s else (old s).1.1) := |
| if h : p.1 then headPoly.congr (by intro s; simp [h]) |
| else tailPoly.congr (by intro s; simp [h]) |
| exact (((endpoint.pair (Poly.const _ natBits (tapeCount q))).comp Poly.natMul).pair |
| (Poly.const _ natBits (cubeEquiv q p.2).val)).comp Poly.natAdd |
| | inr tape => |
| let v := ((verticesPoly (q := q)).pair (Poly.const _ natBits (tapeCount q))).comp Poly.natMul |
| let e := ((indexPoly (q := q)).pair (Poly.const _ natBits (pairTapeCount q))).comp Poly.natMul |
| exact (v.pair ((e.pair (Poly.const _ natBits (pairCubeEquiv q tape).val)).comp Poly.natAdd)).comp Poly.natAdd |
| noncomputable def queryPoly {q : ℕ} (l : Local q) : |
| Poly (code (q := q)) natBits (fun s => |
| queryRank s (Queries.query (GenericGraphTables.relationAt (old s).2) l.1.1 l.1.2)) := by |
| letI := relationFintype q |
| letI : Nonempty (GenericGraphTables.RelationTable q) := ⟨GenericMachine.relationDefault q⟩ |
| exact (((Poly.identity (code (q := q))).pair relationPoly).comp |
| (Poly.finiteBranch code GenericMachine.relationCode natBits (relationCode_injective q) |
| (fun s r => queryRank s (Queries.query (GenericGraphTables.relationAt r) l.1.1 l.1.2)) |
| (fun r => queryRankPoly _))).congr (fun _ => rfl) |
|
|
| def localRow {q : ℕ} (l : Local q) (s : State q) : TableMachine.Row := |
| ((if l.2 then s.1.darts*localCount q + |
| queryRank s (Queries.query (GenericGraphTables.relationAt (old s).2) l.1.1 l.1.2) |
| else s.2*localCount q+(localEventEquiv q l.1.1).val, |
| s.2*blockCount q+(blockEquiv q (l.1,!l.2)).val), |
| Relations.relationFromTable (old s).2 (decide ((old s).1.1=head s)) l.1.1 l.1.2 l.2) |
| noncomputable def localTailPoly {q : ℕ} (l : Local q) : |
| Poly (code (q := q)) natBits (fun s => (localRow l s).1.1) := by |
| if h : l.2 then |
| let count := ((dartsPoly (q := q)).pair (Poly.const _ natBits (localCount q))).comp Poly.natMul |
| exact ((count.pair (queryPoly l)).comp Poly.natAdd).congr (by intro s; simp [localRow,h]) |
| else |
| let offset := ((indexPoly (q := q)).pair (Poly.const _ natBits (localCount q))).comp Poly.natMul |
| exact ((offset.pair (Poly.const _ natBits (localEventEquiv q l.1.1).val)).comp Poly.natAdd).congr |
| (by intro s; simp [localRow,h]) |
| noncomputable def localReversePoly {q : ℕ} (l : Local q) : |
| Poly (code (q := q)) natBits (fun s => (localRow l s).1.2) := |
| ((((indexPoly (q := q)).pair (Poly.const _ natBits (blockCount q))).comp Poly.natMul).pair |
| (Poly.const _ natBits (blockEquiv q (l.1,!l.2)).val)).comp Poly.natAdd |
| noncomputable def localRelationPoly {q : ℕ} (l : Local q) : |
| Poly (code (q := q)) TableMachine.relationCode (fun s => (localRow l s).2) := by |
| letI := relationFintype q |
| exact (relationPoly.pair loopPoly).comp (Poly.finite |
| (prodBits GenericMachine.relationCode boolBits) TableMachine.relationCode |
| (prodBits_injective (relationCode_injective q) boolBits_injective) |
| (fun p => Relations.relationFromTable p.1 p.2 l.1.1 l.1.2 l.2)) |
| noncomputable def localPoly {q : ℕ} (l : Local q) : |
| Poly (code (q := q)) TableMachine.rowCode (localRow l) := |
| ((localTailPoly l).pair (localReversePoly l)).pair (localRelationPoly l) |
|
|
| end VertexCover.Machine.AlphabetMachine |
| end |
|
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| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
