openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionStorageBounds.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionArrayWork | |
| namespace OAI | |
| /-! # Raw compression lists fit the common writable capacity | |
| These bounds apply before the compact-list guard. Hence even a failed | |
| compression test can materialize its raw block lists within the same space. | |
| -/ | |
| namespace LowSpaceSubsetSum.CompressionStorageBounds | |
| variable {n : ℕ} | |
| theorem exponents (hn : 10000 ≤ n) : | |
| compressionHalfSize n ≤ 199*n/1000 ∧ | |
| compressionLeftRawSize n ≤ 199*n/1000 ∧ | |
| ∀ i, compressionRawHalfSize n i ≤ 199*n/1000 := by | |
| have hf := compression_layout_feasible n | |
| rw [compressionHalfSize_eq] at hf ⊢ | |
| unfold compressionLeftRawSize compressionRawHalfSize compressionRightRawSize | |
| rw [compressionHalfSize_eq] | |
| refine ⟨by omega, by omega, ?_⟩ | |
| intro i | |
| split <;> omega | |
| theorem source_lengths (D : CompressionData n) (e : Equiv.Perm (Fin n)) | |
| (hn : 10000 ≤ n) : | |
| (∀ i, ((CompressionArrayBuild.source D e).compact i).length ≤ 199*n/1000) ∧ | |
| (CompressionArrayBuild.source D e).left.length ≤ 199*n/1000 ∧ | |
| ∀ i, ((CompressionArrayBuild.source D e).right i).length ≤ 199*n/1000 := by | |
| have h := exponents hn | |
| refine ⟨?_, ?_, ?_⟩ | |
| · intro i | |
| simpa only [CompressionArrayBuild.source, Finset.length_sort, CompressionData.block, | |
| permuteSubset, Finset.card_map, compressionBlock_card] using h.1 | |
| · simpa only [CompressionArrayBuild.source, Finset.length_sort, D.leftRaw_card] using h.2.1 | |
| · intro i | |
| simpa only [CompressionArrayBuild.source, Finset.length_sort, D.rightRaw_card] using h.2.2 i | |
| theorem raw_bound (a : Fin n → ℤ) (xs : List (Fin n)) | |
| (hx : xs.length ≤ 199*n/1000) : | |
| (CompressionWeights.raw a xs).1.length ≤ compressionDictionaryCapacity n := by | |
| rw [CompressionWeights.raw_length] | |
| exact Nat.pow_le_pow_right (by decide) hx | |
| end LowSpaceSubsetSum.CompressionStorageBounds | |
| end OAI | |