| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.MeasuredPivot |
|
|
| namespace OAI |
|
|
| /-! # Constant-retention binary-search scratch |
|
|
| The search retains one pair of interval endpoints, one midpoint, and the |
| scratch of a single predicate invocation. The peak accumulator does not |
| retain earlier intervals. |
| -/ |
|
|
| namespace LowSpaceSubsetSum.MeasuredPivotCells |
|
|
| variable {n : ℕ} |
|
|
| def step (p : Fin n → (Bool × ℕ) × ℕ) (c : BinaryPivot.Cut n) : |
| (BinaryPivot.Cut n × ℕ) × ℕ := |
| if h : c.lo < c.hi then |
| let mid := (c.lo + c.hi) / 2 |
| have hm : mid < n := by have := c.bounded; dsimp [mid]; omega |
| let q := p ⟨mid, hm⟩ |
| ((if q.1.1 then |
| ⟨mid + 1, c.hi, by dsimp [mid]; omega, c.bounded⟩ |
| else ⟨c.lo, mid, by dsimp [mid]; omega, by have := c.bounded; dsimp [mid]; omega⟩, |
| q.1.2 + 8), q.2 + 24) |
| else ((c, 1), 8) |
|
|
| theorem step_value (p : Fin n → (Bool × ℕ) × ℕ) (c : BinaryPivot.Cut n) : |
| (step p c).1 = BinaryPivot.measuredStep (fun i => (p i).1) c := by |
| unfold step BinaryPivot.measuredStep |
| split <;> rfl |
|
|
| theorem step_cells (p : Fin n → (Bool × ℕ) × ℕ) (R : ℕ) |
| (hp : ∀ i, (p i).2 ≤ R) (c : BinaryPivot.Cut n) : (step p c).2 ≤ R + 24 := by |
| unfold step |
| split |
| · rename_i h |
| exact Nat.add_le_add_right (hp ⟨(c.lo + c.hi) / 2, by have := c.bounded; omega⟩) 24 |
| · omega |
|
|
| def loop (p : Fin n → (Bool × ℕ) × ℕ) : ℕ → BinaryPivot.Cut n → ℕ → ℕ → |
| (BinaryPivot.Cut n × ℕ) × ℕ |
| | 0, c, work, peak => ((c, work), max peak 16) |
| | k + 1, c, work, peak => |
| let q := step p c |
| loop p k q.1.1 (work + q.1.2) (max peak (q.2 + 16)) |
|
|
| theorem loop_value (p : Fin n → (Bool × ℕ) × ℕ) (k : ℕ) |
| (c : BinaryPivot.Cut n) (work peak : ℕ) : |
| (loop p k c work peak).1 = BinaryPivot.measuredRun (fun i => (p i).1) k c work := by |
| induction k generalizing c work peak with |
| | zero => rfl |
| | succ k ih => |
| rw [loop, BinaryPivot.measuredRun, ih, step_value] |
|
|
| theorem loop_cells (p : Fin n → (Bool × ℕ) × ℕ) (R : ℕ) |
| (hp : ∀ i, (p i).2 ≤ R) (k : ℕ) (c : BinaryPivot.Cut n) (work peak : ℕ) : |
| (loop p k c work peak).2 ≤ max peak (R + 40) := by |
| induction k generalizing c work peak with |
| | zero => simp only [loop]; exact max_le_max le_rfl (by omega) |
| | succ k ih => |
| have hs := step_cells p R hp c |
| have hi := ih (step p c).1.1 (work + (step p c).1.2) (max peak ((step p c).2 + 16)) |
| exact hi.trans (by omega) |
|
|
| def search (p : Fin n → (Bool × ℕ) × ℕ) (k : ℕ) := loop p k (BinaryPivot.initial n) 0 0 |
|
|
| theorem search_value (p : Fin n → (Bool × ℕ) × ℕ) (k : ℕ) : |
| (search p k).1 = BinaryPivot.measuredSearch (fun i => (p i).1) k := loop_value _ _ _ _ _ |
|
|
| theorem search_cells (p : Fin n → (Bool × ℕ) × ℕ) (R : ℕ) |
| (hp : ∀ i, (p i).2 ≤ R) (k : ℕ) : (search p k).2 ≤ R + 40 := by |
| simpa only [search, Nat.zero_max] using loop_cells p R hp k (BinaryPivot.initial n) 0 0 |
|
|
| end LowSpaceSubsetSum.MeasuredPivotCells |
|
|
| end OAI |
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|
