| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.CountedReplay |
|
|
| namespace OAI |
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|
| /-! # Cost-carrying bounded tie-buffer construction and lookahead -/ |
|
|
| namespace LowSpaceSubsetSum.CountedFill |
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|
| universe u v |
| variable {α : Type u} {σ : Type v} |
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|
| def fillAcc (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) : ℕ → σ → Array α → ℕ → TieBuffer.Result α σ × ℕ |
| | 0, s, buf, q => (⟨buf, s, q⟩, 1) |
| | fuel + 1, s, buf, q => |
| let fetched := read s |
| match fetched.1 with |
| | none => (⟨buf, s, q + 1⟩, fetched.2 + 5) |
| | some (x, s') => |
| let found := key x |
| if found.1 = K then |
| let rest := fillAcc read key cells K fuel s' (buf.push x) (q + 1) |
| (rest.1, fetched.2 + found.2 + cells x + 8 + rest.2) |
| else (⟨buf, s, q + 1⟩, fetched.2 + found.2 + 6) |
|
|
| theorem fillAcc_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) (fuel : ℕ) (s : σ) (buf : Array α) (q : ℕ) : |
| (fillAcc read key cells K fuel s buf q).1 = |
| TieBuffer.fillAcc (fun s => (read s).1) (fun x => (key x).1) K fuel s buf q := by |
| induction fuel generalizing s buf q with |
| | zero => rfl |
| | succ fuel ih => |
| cases hr : (read s).1 with |
| | none => simp only [fillAcc, TieBuffer.fillAcc, hr] |
| | some z => |
| rcases z with ⟨x, s'⟩ |
| simp only [fillAcc, TieBuffer.fillAcc, hr] |
| split |
| · exact ih _ _ _ |
| · rfl |
|
|
| theorem fillAcc_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) (R L S : ℕ) |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (hs : ∀ x, cells x ≤ S) |
| (fuel : ℕ) (s : σ) (buf : Array α) (q : ℕ) : |
| (fillAcc read key cells K fuel s buf q).2 + (R + L + S + 8) * q ≤ |
| (R + L + S + 8) * (fillAcc read key cells K fuel s buf q).1.queries + 1 := by |
| induction fuel generalizing s buf q with |
| | zero => simp only [fillAcc]; omega |
| | succ fuel ih => |
| have hread := hr s |
| cases hv : (read s).1 with |
| | none => simp only [fillAcc, hv, Nat.mul_add, Nat.mul_one]; omega |
| | some z => |
| rcases z with ⟨x, s'⟩ |
| have hkey := hk x |
| have hcell := hs x |
| simp only [fillAcc, hv] |
| split |
| · have hrest := ih s' (buf.push x) (q + 1) |
| simp only [Nat.mul_add, Nat.mul_one] at hrest |
| dsimp only |
| omega |
| · simp only [Nat.mul_add, Nat.mul_one]; omega |
|
|
| def fill (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) (B : ℕ) (s : σ) : TieBuffer.Result α σ × ℕ := |
| fillAcc read key cells K B s #[] 0 |
|
|
| theorem fill_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) (B : ℕ) (s : σ) : |
| (fill read key cells K B s).1 = |
| TieBuffer.fill (fun s => (read s).1) (fun x => (key x).1) K B s := |
| fillAcc_value _ _ _ _ _ _ _ _ |
|
|
| theorem fill_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) |
| (cells : α → ℕ) (K : ℤ) (R L S : ℕ) |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (hs : ∀ x, cells x ≤ S) |
| (B : ℕ) (s : σ) : |
| (fill read key cells K B s).2 ≤ |
| (R + L + S + 8) * (fill read key cells K B s).1.queries + 1 := by |
| simpa only [fill, Nat.mul_zero, Nat.add_zero] using |
| fillAcc_work read key cells K R L S hr hk hs B s #[] 0 |
|
|
| def hasKey (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) (s : σ) : Bool × ℕ := |
| let fetched := read s |
| match fetched.1 with |
| | none => (false, fetched.2 + 3) |
| | some (x, _) => |
| let found := key x |
| (decide (found.1 = K), fetched.2 + found.2 + 4) |
|
|
| theorem hasKey_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) (s : σ) : |
| (hasKey read key K s).1 = TieBuffer.hasKey (fun s => (read s).1) (fun x => (key x).1) K s := by |
| cases hv : (read s).1 <;> simp [hasKey, TieBuffer.hasKey, hv] |
|
|
| theorem hasKey_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) |
| (R L : ℕ) (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) (s : σ) : |
| (hasKey read key K s).2 ≤ R + L + 4 := by |
| have hread := hr s |
| cases hv : (read s).1 with |
| | none => simp only [hasKey, hv]; omega |
| | some z => |
| rcases z with ⟨x, s'⟩ |
| have hkey := hk x |
| simp only [hasKey, hv] |
| omega |
|
|
| end LowSpaceSubsetSum.CountedFill |
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| end OAI |
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