openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionPairCells.lean
| 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 | |