| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.BufferReplay |
|
|
| namespace OAI |
|
|
| /-! # Cost-carrying tie-suffix replay |
|
|
| Every performed reader, key and callback evaluation contributes its own |
| counter. The result is exactly the bounded replay. The work bound |
| uses its actual stopped query/report counts, rather than the supplied fuel. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.CountedReplay |
|
|
| universe u v w |
| variable {α : Type u} {σ : Type v} {τ : Type w} |
|
|
| def runAcc (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) : |
| ℕ → σ → τ → ℕ → ℕ → TieReplay.Result σ τ × ℕ |
| | 0, s, a, q, r => (⟨s, a, q, r, false, false⟩, 1) |
| | fuel + 1, s, a, q, r => |
| let fetched := read s |
| match fetched.1 with |
| | none => (⟨s, a, q + 1, r, true, false⟩, fetched.2 + 5) |
| | some (x, s') => |
| let found := key x |
| if found.1 = K then |
| let emitted := report x a |
| if emitted.1.2 then |
| (⟨s', emitted.1.1, q + 1, r + 1, false, true⟩, |
| fetched.2 + found.2 + emitted.2 + 8) |
| else |
| let rest := runAcc read key K report fuel s' emitted.1.1 (q + 1) (r + 1) |
| (rest.1, fetched.2 + found.2 + emitted.2 + 8 + rest.2) |
| else (⟨s, a, q + 1, r, true, false⟩, fetched.2 + found.2 + 6) |
|
|
| theorem runAcc_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) : |
| (runAcc read key K report fuel s a q r).1 = |
| TieReplay.runAcc (fun s => (read s).1) (fun x => (key x).1) K |
| (fun x a => (report x a).1) fuel s a q r := by |
| induction fuel generalizing s a q r with |
| | zero => rfl |
| | succ fuel ih => |
| cases hr : (read s).1 with |
| | none => simp only [runAcc, TieReplay.runAcc, hr] |
| | some z => |
| rcases z with ⟨x, s'⟩ |
| simp only [runAcc, TieReplay.runAcc, hr] |
| split |
| · split |
| · rfl |
| · exact ih _ _ _ _ |
| · rfl |
|
|
| theorem runAcc_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (R L C : ℕ) |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) |
| (hc : ∀ x a, (report x a).2 ≤ C) |
| (fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) : |
| (runAcc read key K report fuel s a q r).2 + (R + L + 8) * q + C * r ≤ |
| (R + L + 8) * (runAcc read key K report fuel s a q r).1.queries + |
| C * (runAcc read key K report fuel s a q r).1.reports + 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 |
|
|
| def run (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) : |
| TieReplay.Result σ τ × ℕ := runAcc read key K report fuel s a 0 0 |
|
|
| theorem run_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (fuel : ℕ) (s : σ) (a : τ) : |
| (run read key K report fuel s a).1 = |
| TieReplay.run (fun s => (read s).1) (fun x => (key x).1) K |
| (fun x a => (report x a).1) fuel s a := runAcc_value _ _ _ _ _ _ _ _ _ |
|
|
| theorem run_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (report : α → τ → (τ × Bool) × ℕ) (R L C : ℕ) |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) |
| (hc : ∀ x a, (report x a).2 ≤ C) (fuel : ℕ) (s : σ) (a : τ) : |
| (run read key K report fuel s a).2 ≤ |
| (R + L + 8) * (run read key K report fuel s a).1.queries + |
| C * (run read key K report fuel s a).1.reports + 1 := by |
| simpa only [run, Nat.mul_zero, Nat.add_zero] using |
| runAcc_work read key K report R L C hr hk hc fuel s a 0 0 |
|
|
| end LowSpaceSubsetSum.CountedReplay |
|
|
| end OAI |
|
|
