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import OAI.Computability.VertexCover.Machines.NatDiv
import OAI.Computability.VertexCover.Machines.TableBase
namespace OAI
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namespace VertexCover.Machine
open UniqueGames.Foundations.PCP
namespace ExpandMachine
abbrev Input (q : ℕ) := Σ v, ExpanderTables.Table v q
abbrev Data := ℕ × List ℕ
def dataCode : Data → List Bool := prodBits natBits (listBits natBits)
def erase {q : ℕ} (T : Input q) : Data := (T.1,T.2.rows.toList.map Fin.val)
def code {q : ℕ} (T : Input q) : List Bool := dataCode (erase T)
def lookup (q : ℕ) (p : List ℕ × (ℕ × ℕ)) : ℕ × ℕ :=
let r := p.1.getD (p.2.2+q*p.2.1) 0
(r/q,r%q)
noncomputable def lookupPoly (q : ℕ) :
Poly (prodBits (listBits natBits) (prodBits natBits natBits)) (prodBits natBits natBits) (lookup q) := by
let e := prodBits (listBits natBits) (prodBits natBits natBits)
let rows := Poly.fst (listBits natBits) (prodBits natBits natBits)
let x := Poly.snd (listBits natBits) (prodBits natBits natBits)
let v := x.comp (Poly.fst natBits natBits)
let port := x.comp (Poly.snd natBits natBits)
let base := ((Poly.const e natBits q).pair v).comp Poly.natMul
let idx := (port.pair base).comp Poly.natAdd
let r := (idx.pair rows).comp (Poly.listGetD natBits 0)
let qp := r.pair (Poly.const e natBits q)
exact ((qp.comp Poly.natDiv).pair (qp.comp Poly.natMod)).congr (fun _ => by
simp only [Function.comp_apply,lookup,List.headD_eq_head?_getD,List.head?_drop,List.getD_eq_getElem?_getD])
theorem lookup_erase {q v : ℕ} (T : ExpanderTables.Table v q) (x : Fin v × Fin q) :
lookup q (T.rows.toList.map Fin.val,(x.1.val,x.2.val)) =
((ExpanderTables.lookup T x).1.val,(ExpanderTables.lookup T x).2.val) := by
have hi : x.2.val+q*x.1.val < (T.rows.toList.map Fin.val).length := by
simp only [List.length_map,Vector.length_toList]
exact (ExpanderTables.rowIndex v q x).isLt
rw [lookup,List.getD_eq_getElem _ _ hi,List.getElem_map]
rfl
def stepRow (d : ℕ) (H : List ℕ) (p : Data × ℕ) : ℕ :=
let D := d*d
let g := D*D
let x := p.2/D
let port := p.2%D
let first := lookup d (H,(x%g,port/d))
let outerFirst := lookup D (p.1.2,(x/g,first.1/D))
let outerSecond := lookup D (p.1.2,(outerFirst.1,first.1%D))
let last := lookup d (H,(outerFirst.2+D*outerSecond.2,port%d))
(first.2+d*last.2)+D*(last.1+g*outerSecond.1)
def stepData (d : ℕ) (H : List ℕ) (T : Data) : Data :=
let v := T.1*((d*d)*(d*d))
(v,(List.range (v*(d*d))).map (fun i => stepRow d H (T,i)))
noncomputable def stepRowPoly (d : ℕ) (H : List ℕ) :
Poly (prodBits dataCode natBits) natBits (stepRow d H) := by
let e := prodBits dataCode natBits
let D := d*d
let g := D*D
let c (n : ℕ) := Poly.const e natBits n
let T := Poly.fst dataCode natBits
let rows := T.comp (Poly.snd natBits (listBits natBits))
let i := Poly.snd dataCode natBits
let x := (i.pair (c D)).comp Poly.natDiv
let port := (i.pair (c D)).comp Poly.natMod
let Hc := Poly.const e (listBits natBits) H
let first := (Hc.pair (((x.pair (c g)).comp Poly.natMod).pair
((port.pair (c d)).comp Poly.natDiv))).comp (lookupPoly d)
let firstV := first.comp (Poly.fst natBits natBits)
let firstP := first.comp (Poly.snd natBits natBits)
let out₁ := (rows.pair (((x.pair (c g)).comp Poly.natDiv).pair
((firstV.pair (c D)).comp Poly.natDiv))).comp (lookupPoly D)
let out₁V := out₁.comp (Poly.fst natBits natBits)
let out₁P := out₁.comp (Poly.snd natBits natBits)
let out₂ := (rows.pair (out₁V.pair ((firstV.pair (c D)).comp Poly.natMod))).comp (lookupPoly D)
let out₂V := out₂.comp (Poly.fst natBits natBits)
let out₂P := out₂.comp (Poly.snd natBits natBits)
let lastV := (out₁P.pair (((c D).pair out₂P).comp Poly.natMul)).comp Poly.natAdd
let last := (Hc.pair (lastV.pair ((port.pair (c d)).comp Poly.natMod))).comp (lookupPoly d)
let v := ((last.comp (Poly.fst natBits natBits)).pair
(((c g).pair out₂V).comp Poly.natMul)).comp Poly.natAdd
let p := (firstP.pair (((c d).pair (last.comp (Poly.snd natBits natBits))).comp Poly.natMul)).comp Poly.natAdd
exact (p.pair (((c D).pair v).comp Poly.natMul)).comp Poly.natAdd
noncomputable def stepDataPoly (d : ℕ) (H : List ℕ) : Poly dataCode dataCode (stepData d H) := by
let v := ((Poly.fst natBits (listBits natBits)).pair
(Poly.const dataCode natBits ((d*d)*(d*d)))).comp Poly.natMul
let len := (v.pair (Poly.const dataCode natBits (d*d))).comp Poly.natMul
exact v.pair (Poly.tabulate dataCode natBits (0,[]) 0 len (stepRowPoly d H))
end ExpandMachine
end VertexCover.Machine
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
end OAI
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