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5d5048e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | 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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