openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionDomainCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.IntervalDomainCells | |
| namespace OAI | |
| /-! # Live cells for the ten concrete compression domains | |
| Only arithmetic interval tests, stored permutation lookups and list sorts | |
| are executed. Completed domains are retained while subsequent rows are | |
| constructed, and the source finite sets appear only in the value theorem. | |
| -/ | |
| namespace LowSpaceSubsetSum.CompressionDomainCells | |
| open scoped BigOperators | |
| variable {n : ℕ} | |
| def run (v : Vector (Fin n) n) : (CompressionArrayBuild.Blocks n × ℕ) × ℕ := | |
| let compact := Vector.ofFn (fun i : Fin 7 => | |
| IntervalDomainCells.run v (i.val*compressionHalfSize n) (compressionHalfSize n)) | |
| let left := IntervalDomainCells.run v (7*compressionHalfSize n) (compressionLeftRawSize n) | |
| let right := Vector.ofFn (fun i : Fin 2 => IntervalDomainCells.run v | |
| (IntervalDomainExecution.rightStart n i) (IntervalDomainExecution.rightLength n i)) | |
| ((⟨fun i => compact[i.val].1.1, left.1.1, fun i => right[i.val].1.1⟩, | |
| (∑ i : Fin 7, compact[i.val].1.2)+left.1.2+(∑ i : Fin 2, right[i.val].1.2)+20), | |
| (∑ i : Fin 7, compact[i.val].2)+left.2+(∑ i : Fin 2, right[i.val].2)+40*n+64) | |
| theorem value (v : Vector (Fin n) n) : (run v).1 = CompressionDomains.prepare v := by | |
| rw [← IntervalDomainExecution.compression_source] | |
| simp only [run, IntervalDomainExecution.compression, Vector.getElem_ofFn, | |
| IntervalDomainCells.run_value] | |
| theorem cells (v : Vector (Fin n) n) : (run v).2 ≤ 2048*(n+1)^2 := by | |
| have hc := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 7))) => | |
| IntervalDomainCells.run_cells v (i.val*compressionHalfSize n) (compressionHalfSize n)) | |
| have hr := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 2))) => | |
| IntervalDomainCells.run_cells v | |
| (IntervalDomainExecution.rightStart n i) (IntervalDomainExecution.rightLength n i)) | |
| have hl := IntervalDomainCells.run_cells v (7*compressionHalfSize n) (compressionLeftRawSize n) | |
| simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul] at hc hr | |
| simp only [run, Vector.getElem_ofFn, Fin.eta] | |
| nlinarith only [hc, hr, hl, Nat.zero_le (n^2)] | |
| end LowSpaceSubsetSum.CompressionDomainCells | |
| end OAI | |