File size: 4,486 Bytes
5d5048e | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.CountedReplay
namespace OAI
/-! # Cost-carrying bounded tie-buffer construction and lookahead -/
namespace LowSpaceSubsetSum.CountedFill
universe u v
variable {α : Type u} {σ : Type v}
def fillAcc (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) : ℕ → σ → Array α → ℕ → TieBuffer.Result α σ × ℕ
| 0, s, buf, q => (⟨buf, s, q⟩, 1)
| fuel + 1, s, buf, q =>
let fetched := read s
match fetched.1 with
| none => (⟨buf, s, q + 1⟩, fetched.2 + 5)
| some (x, s') =>
let found := key x
if found.1 = K then
let rest := fillAcc read key cells K fuel s' (buf.push x) (q + 1)
(rest.1, fetched.2 + found.2 + cells x + 8 + rest.2)
else (⟨buf, s, q + 1⟩, fetched.2 + found.2 + 6)
theorem fillAcc_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) (fuel : ℕ) (s : σ) (buf : Array α) (q : ℕ) :
(fillAcc read key cells K fuel s buf q).1 =
TieBuffer.fillAcc (fun s => (read s).1) (fun x => (key x).1) K fuel s buf q := by
induction fuel generalizing s buf q with
| zero => rfl
| succ fuel ih =>
cases hr : (read s).1 with
| none => simp only [fillAcc, TieBuffer.fillAcc, hr]
| some z =>
rcases z with ⟨x, s'⟩
simp only [fillAcc, TieBuffer.fillAcc, hr]
split
· exact ih _ _ _
· rfl
theorem fillAcc_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) (R L S : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (hs : ∀ x, cells x ≤ S)
(fuel : ℕ) (s : σ) (buf : Array α) (q : ℕ) :
(fillAcc read key cells K fuel s buf q).2 + (R + L + S + 8) * q ≤
(R + L + S + 8) * (fillAcc read key cells K fuel s buf q).1.queries + 1 := by
induction fuel generalizing s buf q with
| zero => simp only [fillAcc]; omega
| succ fuel ih =>
have hread := hr s
cases hv : (read s).1 with
| none => simp only [fillAcc, hv, Nat.mul_add, Nat.mul_one]; omega
| some z =>
rcases z with ⟨x, s'⟩
have hkey := hk x
have hcell := hs x
simp only [fillAcc, hv]
split
· have hrest := ih s' (buf.push x) (q + 1)
simp only [Nat.mul_add, Nat.mul_one] at hrest
dsimp only
omega
· simp only [Nat.mul_add, Nat.mul_one]; omega
def fill (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) (B : ℕ) (s : σ) : TieBuffer.Result α σ × ℕ :=
fillAcc read key cells K B s #[] 0
theorem fill_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) (B : ℕ) (s : σ) :
(fill read key cells K B s).1 =
TieBuffer.fill (fun s => (read s).1) (fun x => (key x).1) K B s :=
fillAcc_value _ _ _ _ _ _ _ _
theorem fill_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ)
(cells : α → ℕ) (K : ℤ) (R L S : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (hs : ∀ x, cells x ≤ S)
(B : ℕ) (s : σ) :
(fill read key cells K B s).2 ≤
(R + L + S + 8) * (fill read key cells K B s).1.queries + 1 := by
simpa only [fill, Nat.mul_zero, Nat.add_zero] using
fillAcc_work read key cells K R L S hr hk hs B s #[] 0
def hasKey (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) (s : σ) : Bool × ℕ :=
let fetched := read s
match fetched.1 with
| none => (false, fetched.2 + 3)
| some (x, _) =>
let found := key x
(decide (found.1 = K), fetched.2 + found.2 + 4)
theorem hasKey_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) (s : σ) :
(hasKey read key K s).1 = TieBuffer.hasKey (fun s => (read s).1) (fun x => (key x).1) K s := by
cases hv : (read s).1 <;> simp [hasKey, TieBuffer.hasKey, hv]
theorem hasKey_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(R L : ℕ) (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (s : σ) :
(hasKey read key K s).2 ≤ R + L + 4 := by
have hread := hr s
cases hv : (read s).1 with
| none => simp only [hasKey, hv]; omega
| some z =>
rcases z with ⟨x, s'⟩
have hkey := hk x
simp only [hasKey, hv]
omega
end LowSpaceSubsetSum.CountedFill
end OAI
|