| import Mathlib |
| import OAI.Computability.VertexCover.PCP.PoweringAddresses |
| import OAI.Computability.VertexCover.Machines.Walk |
|
|
| namespace OAI |
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|
| namespace VertexCover.Machine |
|
|
| namespace FixedSearch |
|
|
| def run {α ι β : Type} (p : α → ι → Bool) (f : α → ι → β) (z : β) : List ι → α → β |
| | [], _ => z |
| | i::xs, a => if p a i then f a i else run p f z xs a |
|
|
| noncomputable def poly {α ι β : Type} (ea : α → List Bool) (eb : β → List Bool) |
| (p : α → ι → Bool) (f : α → ι → β) (z : β) |
| (cp : ∀ i, Poly ea boolBits (fun a => p a i)) |
| (cf : ∀ i, Poly ea eb (fun a => f a i)) : |
| (xs : List ι) → Poly ea eb (run p f z xs) |
| | [] => Poly.const ea eb z |
| | i::xs => (cp i).ite (cf i) (poly ea eb p f z cp cf xs) |
|
|
| theorem eq_scan {ι β : Type} (p : ι → Prop) [DecidablePred p] (f : ι → β) (z : β) : |
| ∀ (xs : List ι) (h : ∃ i ∈ xs, p i), |
| run (fun (_ : Unit) i => decide (p i)) (fun _ i => f i) z xs () = |
| f (UniqueGames.Foundations.PCP.PoweringLabels.scanWitness p xs h).val := by |
| intro xs |
| induction xs with |
| | nil => intro h; simp at h |
| | cons i xs ih => |
| intro h |
| by_cases hp : p i |
| · simp [run,UniqueGames.Foundations.PCP.PoweringLabels.scanWitness,hp] |
| · simp only [run,decide_eq_true_eq,hp,↓reduceIte, |
| UniqueGames.Foundations.PCP.PoweringLabels.scanWitness,↓reduceDIte] |
| exact ih _ |
|
|
| end FixedSearch |
|
|
| namespace AddressMachine |
| open UniqueGames.Foundations.PCP |
| open PoweringLabels PoweringAddresses WalkMachine |
|
|
| def search {d : ℕ} (t : ℕ) (a : PaddedLabel (Fin d) t GraphTables.Label) |
| (x : State d × ℕ) : GraphTables.Label := |
| FixedSearch.run (fun x w => decide ((word w.1.val w.2 x.1).2.val=x.2)) |
| (fun _ w => a w) 0 (allAddresses d t) x |
|
|
| noncomputable def searchPoly {d : ℕ} (t : ℕ) (a : PaddedLabel (Fin d) t GraphTables.Label) : |
| Poly (prodBits (code (d := d)) natBits) finCode (search t a) := by |
| exact FixedSearch.poly (prodBits code natBits) finCode _ _ 0 |
| (fun w => ((((Poly.fst code natBits).comp (wordPoly w.1.val w.2)).comp vertexPoly).pair |
| (Poly.snd code natBits)).comp Poly.natEq) |
| (fun w => Poly.const _ finCode (a w)) (allAddresses d t) |
|
|
| theorem search_decode {d : ℕ} (T : PortTables.Input d) (t : ℕ) (v : Fin T.1) |
| (u : Ball (PortTables.portGraph T.2) t v) (a : PaddedLabel (Fin d) t GraphTables.Label) : |
| search t a ((⟨T,v⟩ : State d),u.val.val) = |
| decode (finitePortSelector (PortTables.portGraph T.2) t v) a u := by |
| let p := fun w : PortWords (Fin d) t => |
| (wordToBall (PortTables.portGraph T.2) t v w).val = u.val |
| have h : ∃ w ∈ allAddresses d t, p w := by |
| obtain ⟨w,hw⟩ := wordToBall_surjective (PortTables.portGraph T.2) t v u |
| exact ⟨w,mem_allAddresses d t w,congrArg Subtype.val hw⟩ |
| have hs := FixedSearch.eq_scan p a 0 (allAddresses d t) h |
| change _ = a (scanWitness p (allAddresses d t) h).val |
| rw [← hs] |
| unfold search |
| generalize allAddresses d t = xs |
| induction xs with |
| | nil => rfl |
| | cons w xs ih => |
| dsimp only [FixedSearch.run] |
| have hp : decide ((word w.1.val w.2 (⟨T,v⟩ : State d)).2.val = u.val.val) = decide (p w) := by |
| simp only [p,word,wordToBall,Fin.val_inj] |
| rfl |
| exact if_congr (Iff.of_eq (congrArg (fun c : Bool => c=true) hp)) |
| rfl ih |
|
|
| end AddressMachine |
| end VertexCover.Machine |
| 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 |
|
|
