| import Mathlib |
| import OAI.Computability.VertexCover.Machines.ListFilter |
| import OAI.Computability.VertexCover.Machines.Generic |
|
|
| namespace OAI |
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| namespace VertexCover.Machine.AlphabetMachine |
| open UniqueGames.Foundations.PCP |
| open AlphabetTable |
| open Enumeration |
|
|
| @[instance_reducible] noncomputable def relationFintype (q : ℕ) : Fintype (GenericGraphTables.RelationTable q) := |
| Fintype.ofEquiv (Fin (q*q) → Bool) |
| {toFun := Vector.ofFn, invFun := fun r i => r[i] |
| left_inv := by intro f; funext i; simp |
| right_inv := by intro r; ext i; simp} |
| theorem relationCode_injective (q : ℕ) : Function.Injective (GenericMachine.relationCode (q := q)) := |
| fun _ _ h => Vector.toList_inj.mp h |
|
|
| abbrev State (q : ℕ) := GenericGraphTables.Table q × ℕ |
| def code {q : ℕ} : State q → List Bool := prodBits GenericMachine.code natBits |
| def old {q : ℕ} (s : State q) : GenericMachine.Row q := (GenericMachine.erase s.1).2.getD s.2 (GenericMachine.rowDefault q) |
| def head {q : ℕ} (s : State q) : ℕ := |
| ((GenericMachine.erase s.1).2.getD (old s).1.2 (GenericMachine.rowDefault q)).1.1 |
| noncomputable def tablePoly {q : ℕ} : Poly (code (q := q)) GenericMachine.code Prod.fst := Poly.fst _ _ |
| noncomputable def indexPoly {q : ℕ} : Poly (code (q := q)) natBits Prod.snd := Poly.snd _ _ |
| noncomputable def verticesPoly {q : ℕ} : Poly (code (q := q)) natBits (fun s => s.1.vertices) := |
| tablePoly.comp GenericMachine.verticesPoly |
| noncomputable def dartsPoly {q : ℕ} : Poly (code (q := q)) natBits (fun s => s.1.darts) := |
| tablePoly.comp GenericMachine.dartsPoly |
| noncomputable def oldPoly {q : ℕ} : Poly (code (q := q)) GenericMachine.rowCode old := GenericMachine.lookupPoly |
| noncomputable def tailPoly {q : ℕ} : Poly (code (q := q)) natBits (fun s => (old s).1.1) := |
| oldPoly.comp ((Poly.fst (prodBits natBits natBits) GenericMachine.relationCode).comp (Poly.fst natBits natBits)) |
| noncomputable def reversePoly {q : ℕ} : Poly (code (q := q)) natBits (fun s => (old s).1.2) := |
| oldPoly.comp ((Poly.fst (prodBits natBits natBits) GenericMachine.relationCode).comp (Poly.snd natBits natBits)) |
| noncomputable def relationPoly {q : ℕ} : Poly (code (q := q)) GenericMachine.relationCode (fun s => (old s).2) := |
| oldPoly.comp (Poly.snd (prodBits natBits natBits) GenericMachine.relationCode) |
| noncomputable def headPoly {q : ℕ} : Poly (code (q := q)) natBits head := |
| ((tablePoly.pair reversePoly).comp GenericMachine.lookupPoly).comp |
| ((Poly.fst (prodBits natBits natBits) GenericMachine.relationCode).comp (Poly.fst natBits natBits)) |
| noncomputable def loopPoly {q : ℕ} : Poly (code (q := q)) boolBits (fun s => decide ((old s).1.1=head s)) := |
| (tailPoly.pair headPoly).comp Poly.natEq |
|
|
| abbrev Local (q : ℕ) := (AlphabetGraph.LocalEvent (Fin q) × Fin 6) × Bool |
| def blockCount (q : ℕ) : ℕ := localCount q * 12 |
| def blockEquiv (q : ℕ) : Local q ≃ Fin (blockCount q) := |
| (productEquiv (productEquiv (localEventEquiv q) (Equiv.refl (Fin 6))) boolEquiv).trans |
| (finCongr (by simp [blockCount,Nat.mul_assoc])) |
| theorem block_pos (q : ℕ) : 0<blockCount q := by |
| unfold blockCount localCount fiveTapeCount pairTapeCount tapeCount |
| positivity |
| theorem block_val (q : ℕ) (l : Local q) : |
| (blockEquiv q l).val = (localEventEquiv q l.1.1).val*12 + l.1.2.val*2+bitValue l.2 := by |
| change (boolEquiv l.2).val+2*(l.1.2.val+6*(localEventEquiv q l.1.1).val)=_ |
| rw [boolEquiv_val] |
| omega |
| theorem dart_block_val {q m : ℕ} (e : Fin m) (l : Local q) : |
| (dartEquiv m q (((e,l.1.1),l.1.2),l.2)).val = (blockEquiv q l).val+blockCount q*e.val := by |
| rw [dartEquiv_val,eventEquiv_val,block_val] |
| unfold blockCount |
| ring |
|
|
| end VertexCover.Machine.AlphabetMachine |
| end |
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| end |
| end |
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| end |
| end |
|
|
| end OAI |
|
|
