File size: 3,363 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedOuterReplay
import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedReplayLoops
namespace OAI
namespace LowSpaceSubsetSum.BudgetedOuterLoop
universe u v w
variable {α : Type u} {σ : Type v} {τ : Type w}
def loop (charge : ℕ) (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → ℕ → Option ((τ × Bool) × ℕ) × ℕ) :
ℕ → σ → τ → ℕ → ℕ → ℕ → ℕ → Option (TieReplay.Result σ τ × ℕ) × ℕ
| fuel, s, a, q, r, budget, work =>
if charge ≤ budget then
let left := budget - charge
match fuel with
| 0 => (some (⟨s, a, q, r, false, false⟩, left), work + 1)
| fuel + 1 =>
let fetched := read s
match fetched.1 with
| none => (some (⟨s, a, q + 1, r, true, false⟩, left), work + (fetched.2 + 5))
| some (x, s') =>
let found := key x
if found.1 = K then
let reported := report x a left
match reported.1 with
| none => (none, work + (fetched.2 + found.2 + reported.2 + 8))
| some ((next, halt), remaining) =>
if halt then
(some (⟨s', next, q + 1, r + 1, false, true⟩, remaining),
work + (fetched.2 + found.2 + reported.2 + 8))
else loop charge read key K report fuel s' next (q + 1) (r + 1) remaining
(work + (fetched.2 + found.2 + reported.2 + 8))
else (some (⟨s, a, q + 1, r, true, false⟩, left), work + (fetched.2 + found.2 + 6))
else (none, work + 1)
theorem loop_eq (charge : ℕ) (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → ℕ → Option ((τ × Bool) × ℕ) × ℕ)
(fuel : ℕ) (s : σ) (a : τ) (q r budget work : ℕ) :
loop charge read key K report fuel s a q r budget work =
((BudgetedOuterReplay.execute charge read key K report fuel s a q r budget).1,
work + (BudgetedOuterReplay.execute charge read key K report fuel s a q r budget).2) := by
induction fuel generalizing s a q r budget work with
| zero => rw [loop, BudgetedOuterReplay.execute]; split <;> rfl
| succ fuel ih =>
rw [loop, BudgetedOuterReplay.execute]
split
· dsimp only
cases hr : (read s).1 with
| none => dsimp only
| some p =>
rcases p with ⟨x, s'⟩
dsimp only
split
· cases hc : (report x a (budget - charge)).1 with
| none => dsimp only
| some p =>
rcases p with ⟨⟨next, halt⟩, remaining⟩
dsimp only
split
· rfl
· rw [ih]
simp only [Nat.add_assoc]
· rfl
· rfl
def execute (charge : ℕ) (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ)
(report : α → τ → ℕ → Option ((τ × Bool) × ℕ) × ℕ)
(fuel : ℕ) (s : σ) (a : τ) (q r budget : ℕ) :=
loop charge read key K report fuel s a q r budget 0
@[csimp] theorem execute_eq : @BudgetedOuterReplay.execute.{u, v, w} = @execute.{u, v, w} := by
funext α σ τ charge read key K report fuel s a q r budget
rw [execute, loop_eq]
simp only [Nat.zero_add]
end LowSpaceSubsetSum.BudgetedOuterLoop
end OAI
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