File size: 3,209 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedStoppedJoin
namespace OAI
/-! # Tail-recursive execution of the prepaid stopped join
The loop keeps the completed-step and measured-work counters in its state.
No frame containing a prior cursor survives a recursive call.
-/
namespace LowSpaceSubsetSum.BudgetedStoppedLoop
universe s t w
variable {σ₁ : Type s} {σ₂ : Type t} {τ : Type w}
local notation "State" => StreamJoin.State σ₁ σ₂ τ
local notation "Result" => StreamJoin.StoppedResult σ₁ σ₂ τ
def shift (steps : ℕ) (out : Result) : Result := ⟨out.state, out.steps + steps⟩
def loop (step : State → ℕ → Option (State × ℕ) × ℕ) :
ℕ → State → ℕ → ℕ → ℕ → Option (Result × ℕ) × ℕ
| fuel, st, steps, budget, work =>
if 4 ≤ budget then
match fuel with
| 0 => (some (⟨st, steps⟩, budget - 4), work + 1)
| fuel + 1 =>
if st.done then (some (⟨st, steps⟩, budget - 4), work + 1) else
let next := step st (budget - 4)
match next.1 with
| none => (none, work + next.2 + 2)
| some (state, left) => loop step fuel state (steps + 1) left (work + next.2 + 3)
else (none, work + 1)
theorem loop_eq (step : State → ℕ → Option (State × ℕ) × ℕ)
(fuel : ℕ) (st : State) (steps budget work : ℕ) :
loop step fuel st steps budget work =
((BudgetedStoppedJoin.execute step fuel st budget).1.map (fun (out, left) => (shift steps out, left)),
work + (BudgetedStoppedJoin.execute step fuel st budget).2) := by
induction fuel generalizing st steps budget work with
| zero =>
rw [loop, BudgetedStoppedJoin.execute]
split <;> simp only [Option.map_some, Option.map_none, shift, Nat.zero_add]
| succ fuel ih =>
rw [loop, BudgetedStoppedJoin.execute]
split
next hb =>
by_cases hd : st.done = true
· simp only [hd, ↓reduceIte, Option.map_some, shift, Nat.zero_add]
· simp only [hd, Bool.false_eq_true, ↓reduceIte]
cases hn : (step st (budget - 4)).1 with
| none => simp only [Option.map_none, Nat.add_assoc]
| some p =>
rcases p with ⟨next, remaining⟩
dsimp only
rw [ih]
simp only [Option.map_map, Function.comp_def, shift, Nat.add_assoc]
apply Prod.ext
· apply congrArg (fun f => (BudgetedStoppedJoin.execute step fuel next remaining).1.map f)
funext p
rcases p with ⟨out, left⟩
simp only
congr 2
omega
· omega
next _ => rfl
def run (step : State → ℕ → Option (State × ℕ) × ℕ)
(fuel : ℕ) (st : State) (budget : ℕ) := loop step fuel st 0 budget 0
theorem run_eq (step : State → ℕ → Option (State × ℕ) × ℕ)
(fuel : ℕ) (st : State) (budget : ℕ) :
run step fuel st budget = BudgetedStoppedJoin.execute step fuel st budget := by
rw [run, loop_eq]
simp only [shift, Nat.add_zero, Nat.zero_add]
change ((BudgetedStoppedJoin.execute step fuel st budget).1.map id, _) = _
apply Prod.ext
· exact Option.map_id'
· rfl
end LowSpaceSubsetSum.BudgetedStoppedLoop
end OAI
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