File size: 2,646 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 81 82 83 84 | import OAI.Computability.SpaceSimulation.Register.Arithmetic
namespace OAI
namespace Fifths
namespace Scan
def all (p : ℕ → Bool) : ℕ → Bool
| 0 => true
| n+1 => all p n && p n
theorem all_iff (p : ℕ → Bool) (n) : all p n = true ↔ ∀ j < n, p j = true := by
induction n with
| zero => simp [all]
| succ n ih =>
rw [all, Bool.and_eq_true, ih]
constructor
· rintro ⟨a,b⟩ j l
by_cases e : j < n
· exact a _ e
· rw [show j = n by omega]
exact b
· exact fun a => ⟨fun j e => a _ (by omega), a n (by omega)⟩
def pick (p : ℕ → Bool) : ℕ → ℕ
| 0 => 0
| n+1 => if p n then n else pick p n
theorem pick_le (p : ℕ → Bool) (n) : pick p n ≤ n := by
induction n with
| zero => rfl
| succ n ih => simp only [pick]; split <;> omega
theorem pick_spec (p : ℕ → Bool) (n) (e : ∃ j < n, p j = true) :
pick p n < n ∧ p (pick p n) = true := by
induction n with
| zero => simp at e
| succ n ih =>
unfold pick
split
next h => exact ⟨by omega,h⟩
next h =>
suffices g : ∃ j < n, p j = true from ⟨by have := (ih g).1; omega,(ih g).2⟩
obtain ⟨j,a,b⟩ := e
refine ⟨j,?_,b⟩
by_contra! v
have l : j = n := by omega
subst j
contradiction
end Scan
namespace Comp
variable {X I : Type} {x : I → List X} {S a b : I → ℕ}
theorem scanAll (p : I → ℕ → Bool) (q : Comp x S a b)
(h : ∀ e : Before b → ℕ, Comp (fun i => x i.1.1) (fun i => S i.1.1) e
(fun i => a i.1.1) → Comp (fun i => x i.1.1) (fun i => S i.1.1) e (fun i => i.1.2)
→ Comp (fun i => x i.1.1) (fun i => S i.1.1) e
(fun i => (p i.1.1 i.1.2).toNat)) :
Comp x S a (fun i => (Scan.all (p i) (b i)).toNat) := by
apply fold (fun i j => (Scan.all (p i) j).toNat) q
(Comp.constant q.source 1)
· exact (Fits.const _ 1).mono (fun _ => Bool.toNat_le _)
· intro e i j r
exact r.both (h e i j)
theorem scanPick (p : I → ℕ → Bool) (q : Comp x S a b)
(h : ∀ e : Before b → ℕ, Comp (fun i => x i.1.1) (fun i => S i.1.1) e
(fun i => a i.1.1) → Comp (fun i => x i.1.1) (fun i => S i.1.1) e (fun i => i.1.2)
→ Comp (fun i => x i.1.1) (fun i => S i.1.1) e
(fun i => (p i.1.1 i.1.2).toNat)) :
Comp x S a (fun i => Scan.pick (p i) (b i)) := by
apply fold (fun i j => Scan.pick (p i) j) q
(Comp.constant q.source 0)
· let f : Through b → I := fun i => i.1.1
exact (q.target.pull f).mono (fun i => (Scan.pick_le _ _).trans i.2)
· intro e i j r
exact (h e i j).iteB j r
end Comp
end Fifths
end OAI
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