File size: 2,926 Bytes
472a116 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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
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