| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.MeasuredFilterCells |
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.BudgetedSeparatorNode |
|
|
| namespace OAI |
|
|
| /-! # The prepaid filter node realizes its local cell allowance -/ |
|
|
| namespace LowSpaceSubsetSum.BudgetedNodeCells |
|
|
| variable {α β : Type*} |
|
|
| def prepare (test₁ : α → ℕ → ℕ → (Bool × ℕ) × ℕ) |
| (test₂ : β → ℕ → ℕ → (Bool × ℕ) × ℕ) |
| (lo hi : ℕ) (xs : List α) (ys : List β) : |
| (BudgetedSeparatorNode.Frame α β × ℕ) × ℕ := |
| let a := MeasuredFilterCells.run (fun x => test₁ x lo hi) xs |
| let b := MeasuredFilterCells.run (fun y => test₂ y lo hi) ys |
| let nx := a.1.1.length |
| let ny := b.1.1.length |
| ((⟨a.1.1, b.1.1, nx, ny, rfl, rfl⟩, a.1.2 + b.1.2 + 2 * (nx + ny) + 12), |
| xs.length + ys.length + 4 + max a.2 (nx + b.2)) |
|
|
| theorem prepare_value (test₁ : α → ℕ → ℕ → (Bool × ℕ) × ℕ) |
| (test₂ : β → ℕ → ℕ → (Bool × ℕ) × ℕ) |
| (lo hi : ℕ) (xs : List α) (ys : List β) : |
| (prepare test₁ test₂ lo hi xs ys).1 = |
| BudgetedSeparatorNode.prepare (fun x l h => (test₁ x l h).1) |
| (fun y l h => (test₂ y l h).1) lo hi xs ys := by |
| simp only [prepare, MeasuredFilterCells.run_value, BudgetedSeparatorNode.prepare, |
| MeasuredSeparatorTraversal.keep] |
|
|
| theorem prepare_cells (test₁ : α → ℕ → ℕ → (Bool × ℕ) × ℕ) |
| (test₂ : β → ℕ → ℕ → (Bool × ℕ) × ℕ) (H : ℕ) |
| (h₁ : ∀ x l h, (test₁ x l h).2 ≤ H) (h₂ : ∀ y l h, (test₂ y l h).2 ≤ H) |
| (lo hi : ℕ) (xs : List α) (ys : List β) : |
| (prepare test₁ test₂ lo hi xs ys).2 ≤ 8 * (xs.length + ys.length + 1) + H := by |
| have ha := MeasuredFilterCells.run_cells (fun x => test₁ x lo hi) H (fun x => h₁ x lo hi) xs |
| have hb := MeasuredFilterCells.run_cells (fun y => test₂ y lo hi) H (fun y => h₂ y lo hi) ys |
| have hl := MeasuredFilterCells.run_length (fun x => test₁ x lo hi) xs |
| have ha' := ha.trans (show 2 * xs.length + H + 4 ≤ |
| 2 * (xs.length + ys.length) + H + 4 by omega) |
| have hb' : (MeasuredFilterCells.run (fun x => test₁ x lo hi) xs).1.1.length + |
| (MeasuredFilterCells.run (fun y => test₂ y lo hi) ys).2 ≤ |
| 2 * (xs.length + ys.length) + H + 4 := by omega |
| have hm := max_le ha' hb' |
| change xs.length + ys.length + 4 + max _ _ ≤ _ |
| omega |
|
|
| end LowSpaceSubsetSum.BudgetedNodeCells |
|
|
| end OAI |
|
|
