| import OAI.Computability.SpaceSimulation.OneTape.Control |
| import OAI.Computability.SpaceSimulation.OneTape.Sweeps |
| import OAI.Computability.SpaceSimulation.Epoch.Theorem |
|
|
| namespace OAI |
|
|
| namespace Fifths.Single |
| open Square Entry Fifths.Comp Radix Tour |
| variable {I X : Type} |
|
|
| namespace Kit |
| variable {M : Single X} {A : M.Sym} {x : I → List X} {S a b j : I → ℕ} {W : I → Site} |
| variable (p : M.Kit A x S a W) |
| include p |
|
|
| theorem onPaper (h : Comp x S a b) : |
| Comp x S a (fun i => (M.tour (x i) (W i)).tape (b i)) := by |
| let d (i : I) := M.glimpse (W i) (M.sensor (x i) none) (b i) |
| let n i := (W i).b |
| apply digits p.ofRadix p.ofBlock (p.subGranted (fun i => Nat.le_succ (n i))) |
| (fun i => M.rpos _) d (fun i j _ => M.sym_lt _ _) |
| intro e g w |
| let k : Before n → I := fun i => i.1.1 |
| let u (i : Before n) := (W (k i)).pin (b (k i)) i.1.2 |
| have c := p.changeInput k g |
| have l := c.ofPosition |
| have v := ((g.trans (h.pull k)).mul c.ofBlock).add w |
| have z := c.double |
| have y := c.read none (u := u) (z.min (v.sub l)) |
| refine (((l.le v).both (v.le (l.add z))).iteB y |
| (Comp.constant v.source (Entry.code (Entry.parse M.A 0)))).congr ?_ |
| intro i |
| change _ = d (k i) _ |
| dsimp only [d,glimpse,Site.reach] |
| simp only [Bool.and_eq_true,decide_eq_true_eq] |
| split <;> rfl |
| |
| theorem runnability : Runnable (fun i => M.tour (x i) (W i)) x S a := |
| { kOf := p.ofK |
| cOf := p.ofC |
| nOf := p.ofBlocks |
| fitReg := p.budget |
| initOf := p.start |
| tapeOf := fun {_U} r {_s _t} o h => (p.changeInput r o).onPaper h |
| selectOf := fun {_U} r {_s _t} o h _ => (p.changeInput r o).selecting h |
| visitOf := fun {_U} r {_s _t _u} o h g a b => (p.changeInput r o).sweeping g h b a } |
| end Kit |
|
|
| noncomputable def lastWord (M : Single X) (x : List X) (W : Site) : ℕ := |
| let D := M.tour x W |
| (D.story D.visits).ctrl |
|
|
| noncomputable def complete (M : Single X) (x : List X) (W : Site) := |
| M.doneAt W (M.lastWord x W) |
| noncomputable def readout (M : Single X) (x : List X) (W : Site) := |
| M.predict W (M.lastWord x W) |
| theorem if_complete (M : Single X) (x : List X) (W : Site) |
| (h : M.complete x W) : M.readout x W = M.outcome x W.T := |
| M.reported x W h |
| theorem some_complete (M : Single X) (x : List X) (T k : ℕ) (h : 0<k) (l : T ≤ k^5) : |
| ∃ a : Fin (k*k), M.complete x (Site.mk T k h a) = true := by |
| exact M.sufficient x T k h l |
|
|
| namespace Kit |
| variable {M : Single X} {A : M.Sym} {x : I → List X} {S a : I → ℕ} {W : I → Site} |
| variable (p : M.Kit A x S a W) |
|
|
| include p |
| theorem solves (b : Scaled S (fun i => |
| let D := M.tour (x i) (W i); D.E * D.per)) : |
| Comp x S a (fun i => (M.complete (x i) (W i)).toNat) |
| ∧ Tells x S a (fun i => M.readout (x i) (W i)) := by |
| have c := Tour.follows p.runnability (fun i => M.tour_bounded (x i) (W i)) b |
| have d := p.parsed c |
| exact ⟨d.time.eq p.ofCap, finite_map (fun y => code (M.finish y)) d.state⟩ |
| end Kit |
| end Fifths.Single |
|
|
| end OAI |
|
|
