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 OAI