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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 | 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
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