| 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 |
|
|
