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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Streams.BufferReplay
namespace OAI
/-! # Cost-carrying tie-suffix replay
Every performed reader, key and callback evaluation contributes its own
counter. The result is exactly the bounded replay. The work bound
uses its actual stopped query/report counts, rather than the supplied fuel.
-/
namespace LowSpaceSubsetSum.CountedReplay
universe u v w
variable {α : Type u} {σ : Type v} {τ : Type w}
def runAcc (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) :
ℕ → σ → τ → ℕ → ℕ → TieReplay.Result σ τ × ℕ
| 0, s, a, q, r => (⟨s, a, q, r, false, false⟩, 1)
| fuel + 1, s, a, q, r =>
let fetched := read s
match fetched.1 with
| none => (⟨s, a, q + 1, r, true, false⟩, fetched.2 + 5)
| some (x, s') =>
let found := key x
if found.1 = K then
let emitted := report x a
if emitted.1.2 then
(⟨s', emitted.1.1, q + 1, r + 1, false, true⟩,
fetched.2 + found.2 + emitted.2 + 8)
else
let rest := runAcc read key K report fuel s' emitted.1.1 (q + 1) (r + 1)
(rest.1, fetched.2 + found.2 + emitted.2 + 8 + rest.2)
else (⟨s, a, q + 1, r, true, false⟩, fetched.2 + found.2 + 6)
theorem runAcc_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) :
(runAcc read key K report fuel s a q r).1 =
TieReplay.runAcc (fun s => (read s).1) (fun x => (key x).1) K
(fun x a => (report x a).1) fuel s a q r := by
induction fuel generalizing s a q r with
| zero => rfl
| succ fuel ih =>
cases hr : (read s).1 with
| none => simp only [runAcc, TieReplay.runAcc, hr]
| some z =>
rcases z with ⟨x, s'⟩
simp only [runAcc, TieReplay.runAcc, hr]
split
· split
· rfl
· exact ih _ _ _ _
· rfl
theorem runAcc_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (R L C : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L)
(hc : ∀ x a, (report x a).2 ≤ C)
(fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) :
(runAcc read key K report fuel s a q r).2 + (R + L + 8) * q + C * r ≤
(R + L + 8) * (runAcc read key K report fuel s a q r).1.queries +
C * (runAcc read key K report fuel s a q r).1.reports + 1 := by
induction fuel generalizing s a q r with
| zero => simp only [runAcc]; omega
| succ fuel ih =>
have hread := hr s
cases hv : (read s).1 with
| none =>
simp only [runAcc, hv, Nat.mul_add, Nat.mul_one]
omega
| some z =>
rcases z with ⟨x, s'⟩
have hkey := hk x
simp only [runAcc, hv]
split
· have hcallback := hc x a
split
· simp only [Nat.mul_add, Nat.mul_one]; omega
· have hrest := ih s' (report x a).1.1 (q + 1) (r + 1)
simp only [Nat.mul_add, Nat.mul_one] at hrest
dsimp only
omega
· simp only [Nat.mul_add, Nat.mul_one]; omega
def run (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) :
TieReplay.Result σ τ × ℕ := runAcc read key K report fuel s a 0 0
theorem run_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) :
(run read key K report fuel s a).1 =
TieReplay.run (fun s => (read s).1) (fun x => (key x).1) K
(fun x a => (report x a).1) fuel s a := runAcc_value _ _ _ _ _ _ _ _ _
theorem run_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (R L C : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L)
(hc : ∀ x a, (report x a).2 ≤ C) (fuel : ℕ) (s : σ) (a : τ) :
(run read key K report fuel s a).2 ≤
(R + L + 8) * (run read key K report fuel s a).1.queries +
C * (run read key K report fuel s a).1.reports + 1 := by
simpa only [run, Nat.mul_zero, Nat.add_zero] using
runAcc_work read key K report R L C hr hk hc fuel s a 0 0
end LowSpaceSubsetSum.CountedReplay
end OAI
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