| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedOuterLoop |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| The callback may exhaust its remaining allowance. Its scratch trace is added |
| to the retained reader, current record and user state before the next tail |
| iteration, and failure drops all local continuations. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.BudgetedOuterCells |
|
|
| universe u v w |
| variable {α : Type u} {σ : Type v} {τ : Type w} |
|
|
| def loop (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) : |
| ℕ → σ → τ → ℕ → ℕ → ℕ → ℕ → ℕ → |
| (Option (TieReplay.Result σ τ × ℕ) × ℕ) × ℕ |
| | fuel, s, a, q, r, budget, work, peak => |
| let live := max peak (stateCells s + userCells a + 16) |
| if charge ≤ budget then |
| let left := budget - charge |
| match fuel with |
| | 0 => ((some (⟨s, a, q, r, false, false⟩, left), work + 1), live) |
| | fuel + 1 => |
| let fetched := read s |
| let live := max live (stateCells s + userCells a + fetched.2 + 16) |
| match fetched.1.1 with |
| | none => ((some (⟨s, a, q + 1, r, true, false⟩, left), work + (fetched.1.2 + 5)), live) |
| | some (x, s') => |
| let found := key x |
| let live := max live |
| (stateCells s + stateCells s' + userCells a + recordCells x + found.2 + 20) |
| if found.1.1 = K then |
| let out := report x a left |
| let live := max live (stateCells s + stateCells s' + userCells a + |
| recordCells x + out.2 + 24) |
| match out.1.1 with |
| | none => ((none, work + (fetched.1.2 + found.1.2 + out.1.2 + 8)), live) |
| | some ((next, halt), remaining) => |
| let live := max live (stateCells s + stateCells s' + userCells a + |
| userCells next + recordCells x + 24) |
| if halt then |
| ((some (⟨s', next, q + 1, r + 1, false, true⟩, remaining), |
| work + (fetched.1.2 + found.1.2 + out.1.2 + 8)), live) |
| else loop charge read key K report stateCells userCells recordCells fuel s' next |
| (q + 1) (r + 1) remaining (work + (fetched.1.2 + found.1.2 + out.1.2 + 8)) live |
| else ((some (⟨s, a, q + 1, r, true, false⟩, left), |
| work + (fetched.1.2 + found.1.2 + 6)), live) |
| else ((none, work + 1), live) |
|
|
| theorem loop_value (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) |
| (fuel : ℕ) (s : σ) (a : τ) (q r budget work peak : ℕ) : |
| (loop charge read key K report stateCells userCells recordCells fuel s a q r budget work peak).1 = |
| BudgetedOuterLoop.loop charge (fun s => (read s).1) (fun x => (key x).1) K |
| (fun x a b => (report x a b).1) fuel s a q r budget work := by |
| induction fuel generalizing s a q r budget work peak with |
| | zero => rw [loop, BudgetedOuterLoop.loop]; split <;> rfl |
| | succ fuel ih => |
| rw [loop, BudgetedOuterLoop.loop] |
| split |
| · dsimp only |
| cases hr : (read s).1.1 with |
| | none => rfl |
| | some p => |
| rcases p with ⟨x, s'⟩ |
| dsimp only |
| split |
| · cases hp : (report x a (budget - charge)).1.1 with |
| | none => rfl |
| | some p => |
| rcases p with ⟨⟨next, halt⟩, remaining⟩ |
| dsimp only |
| split |
| · rfl |
| · exact ih _ _ _ _ _ _ _ |
| · rfl |
| · rfl |
|
|
| theorem loop_cells (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) |
| (S U X R L H : ℕ) (hs : ∀ s, stateCells s ≤ S) (hu : ∀ a, userCells a ≤ U) |
| (hx : ∀ x, recordCells x ≤ X) (hr : ∀ s, (read s).2 ≤ R) |
| (hk : ∀ x, (key x).2 ≤ L) (hp : ∀ x a b, (report x a b).2 ≤ H) |
| (fuel : ℕ) (s : σ) (a : τ) (q r budget work peak : ℕ) : |
| (loop charge read key K report stateCells userCells recordCells fuel s a q r budget work peak).2 ≤ |
| max peak (2 * S + 2 * U + X + R + L + H + 64) := by |
| induction fuel generalizing s a q r budget work peak with |
| | zero => |
| have hs' := hs s |
| have hu' := hu a |
| rw [loop] |
| split <;> dsimp only <;> exact max_le_max le_rfl (by omega) |
| | succ fuel ih => |
| have hs' := hs s |
| have hu' := hu a |
| have hr' := hr s |
| rw [loop] |
| split |
| · dsimp only |
| cases he : (read s).1.1 with |
| | none => dsimp only; omega |
| | some p => |
| rcases p with ⟨x, s'⟩ |
| have hs'' := hs s' |
| have hx' := hx x |
| have hk' := hk x |
| dsimp only |
| split |
| · have hp' := hp x a (budget - charge) |
| cases ho : (report x a (budget - charge)).1.1 with |
| | none => dsimp only; omega |
| | some p => |
| rcases p with ⟨⟨next, halt⟩, remaining⟩ |
| have hu'' := hu next |
| dsimp only |
| split |
| · dsimp only; omega |
| · have hi := ih s' next (q + 1) (r + 1) remaining |
| (work + ((read s).1.2 + (key x).1.2 + (report x a (budget - charge)).1.2 + 8)) |
| (max (max (max (max (max peak (stateCells s + userCells a + 16)) |
| (stateCells s + userCells a + (read s).2 + 16)) |
| (stateCells s + stateCells s' + userCells a + recordCells x + (key x).2 + 20)) |
| (stateCells s + stateCells s' + userCells a + recordCells x + |
| (report x a (budget - charge)).2 + 24)) |
| (stateCells s + stateCells s' + userCells a + userCells next + recordCells x + 24)) |
| exact hi.trans (by omega) |
| · dsimp only; omega |
| · dsimp only |
| exact max_le_max le_rfl (by omega) |
|
|
| def run (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) |
| (fuel : ℕ) (s : σ) (a : τ) (q r budget : ℕ) := |
| loop charge read key K report stateCells userCells recordCells fuel s a q r budget 0 0 |
|
|
| theorem run_value (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) |
| (fuel : ℕ) (s : σ) (a : τ) (q r budget : ℕ) : |
| (run charge read key K report stateCells userCells recordCells fuel s a q r budget).1 = |
| BudgetedOuterReplay.execute charge (fun s => (read s).1) (fun x => (key x).1) K |
| (fun x a b => (report x a b).1) fuel s a q r budget := by |
| rw [run, loop_value, BudgetedOuterLoop.loop_eq] |
| simp only [Nat.zero_add] |
|
|
| theorem run_cells (charge : ℕ) (read : σ → (Option (α × σ) × ℕ) × ℕ) |
| (key : α → (ℤ × ℕ) × ℕ) (K : ℤ) |
| (report : α → τ → ℕ → (Option ((τ × Bool) × ℕ) × ℕ) × ℕ) |
| (stateCells : σ → ℕ) (userCells : τ → ℕ) (recordCells : α → ℕ) |
| (S U X R L H : ℕ) (hs : ∀ s, stateCells s ≤ S) (hu : ∀ a, userCells a ≤ U) |
| (hx : ∀ x, recordCells x ≤ X) (hr : ∀ s, (read s).2 ≤ R) |
| (hk : ∀ x, (key x).2 ≤ L) (hp : ∀ x a b, (report x a b).2 ≤ H) |
| (fuel : ℕ) (s : σ) (a : τ) (q r budget : ℕ) : |
| (run charge read key K report stateCells userCells recordCells fuel s a q r budget).2 ≤ |
| 2 * S + 2 * U + X + R + L + H + 64 := by |
| simpa only [run, Nat.zero_max] using |
| loop_cells charge read key K report stateCells userCells recordCells S U X R L H |
| hs hu hx hr hk hp fuel s a q r budget 0 0 |
|
|
| end LowSpaceSubsetSum.BudgetedOuterCells |
|
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| end OAI |
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