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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 | import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionArrayBuild
namespace OAI
/-! # Live cells for the two materialized compression pair lists
The recursive tail, current mapped row, append frames and output list are
all included. Repeated weights remain distinct occurrences.
-/
namespace LowSpaceSubsetSum.CompressionPairCells
def run : List ℤ → List ℤ → (List ℤ × ℕ) × ℕ
| [], ys => (([], 1), 4*(ys.length+1))
| x :: xs, ys =>
let rest := run xs ys
let row := ys.map (x+·)
let out := row++rest.1.1
((out, rest.1.2+3*ys.length+2),
max (rest.2+8) (4*(out.length+ys.length+1)+8))
theorem value (xs ys : List ℤ) : (run xs ys).1 = CompressionArrayBuild.pairSums xs ys := by
induction xs with
| nil => rfl
| cons x xs ih => simp only [run, CompressionArrayBuild.pairSums, ih]
theorem length (xs ys : List ℤ) : (run xs ys).1.1.length = xs.length*ys.length := by
rw [value, CompressionArrayBuild.pairSums_value, pairSumList_length]
theorem cells (xs ys : List ℤ) :
(run xs ys).2 ≤ 16*(xs.length*ys.length+xs.length+ys.length+1) := by
induction xs with
| nil => simp only [run, List.length_nil, Nat.zero_mul, Nat.zero_add]; omega
| cons x xs ih =>
simp only [run, List.length_cons, List.length_append, List.length_map, length]
apply max_le
· nlinarith only [ih]
· nlinarith only [Nat.zero_le (xs.length*ys.length)]
end LowSpaceSubsetSum.CompressionPairCells
end OAI
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