File size: 2,506 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Counting.CountedReplay
namespace OAI
/-! # Telescoping callback costs through a stopped replay
The callback may itself contain a replay. A potential on its returned state
charges that actual inner work; a uniform worst-case inner fuel bound is not
multiplied by the outer stream length.
-/
namespace LowSpaceSubsetSum.CountedReplay
universe u v w
variable {α : Type u} {σ : Type v} {τ : Type w}
theorem runAcc_work_potential
(read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (potential : τ → ℕ) (R L C : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L)
(hc : ∀ x a, (report x a).2 + potential a ≤ potential (report x a).1.1 + C)
(fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) :
(runAcc read key K report fuel s a q r).2 + (R + L + C + 8) * q + potential a ≤
(R + L + C + 8) * (runAcc read key K report fuel s a q r).1.queries +
potential (runAcc read key K report fuel s a q r).1.state + 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
theorem run_work_potential
(read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → (τ × Bool) × ℕ) (potential : τ → ℕ) (R L C : ℕ)
(hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L)
(hc : ∀ x a, (report x a).2 + potential a ≤ potential (report x a).1.1 + C)
(fuel : ℕ) (s : σ) (a : τ) :
(run read key K report fuel s a).2 + potential a ≤
(R + L + C + 8) * (run read key K report fuel s a).1.queries +
potential (run read key K report fuel s a).1.state + 1 := by
simpa only [run, Nat.mul_zero, Nat.add_zero, Nat.zero_add] using
runAcc_work_potential read key K report potential R L C hr hk hc fuel s a 0 0
end LowSpaceSubsetSum.CountedReplay
end OAI
|