openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Word /WordSeedCompressionExecution.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPermutationExecution | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordSeedPrimeInitialization | |
| namespace OAI | |
| /-! # Actual word-driven compression sampler | |
| The permutation generator retains its range candidate buffers. The prime | |
| width is computed by one multiplication and one division before its | |
| modulus and bounded rejection routine. Exhaustion remains `none`. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.WordSeedMeasured | |
| private def compressionCore {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n) | |
| (w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) : | |
| Execution (WordSeedRealization.compression D hn w hp hq) := | |
| let ell := 249*n/1000 | |
| bind (permutation w n n hp) (fun _ => CompressionSetup.PrimeSpace n) | |
| (fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n) | |
| (fun _ => WordSeedRealization.prime w n ell (by omega) hq) | |
| (fun _ => primeInitialized w n ell (by omega) hq) | |
| def compression {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n) | |
| (w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) : | |
| Execution (WordSeedRealization.compression D hn w hp hq) where | |
| run stream cursor := | |
| let out := (compressionCore D hn w hp hq).run stream cursor | |
| ⟨out.accepted,out.cursor,out.work+6,max 8 (out.cells+8),out.retained+1⟩ | |
| value stream cursor := (compressionCore D hn w hp hq).value stream cursor | |
| def compressionWork (n : ℕ) : ℕ := | |
| n*(13*n^3+5*n+64)+ | |
| (n^3*(12*2^((compressionPrimeExponent n+1)/2)+ | |
| 12*(compressionPrimeExponent n+1)+11)+3*compressionPrimeExponent n+17)+9 | |
| theorem compression_bounded {n : ℕ} (D : CompressionData n) (hn : 168000000 ≤ n) | |
| (w : ℕ) (hp : Nat.clog 2 n ≤ w) (hq : compressionPrimeExponent n ≤ w) : | |
| (compression D hn w hp hq).Bounded (compressionWork n) | |
| ((n+3)*(2*n^3+256)) ((n+2)*(n^3+64)) := by | |
| have hh := bind_bounded (permutation w n n hp) | |
| (fun _ => CompressionSetup.PrimeSpace n) | |
| (fun _ => primeCoupling (by unfold compressionPrimeExponent; omega) n) | |
| (fun _ => WordSeedRealization.prime w n (compressionPrimeExponent n) | |
| (by unfold compressionPrimeExponent; omega) hq) | |
| (fun _ => primeInitialized w n (compressionPrimeExponent n) | |
| (by unfold compressionPrimeExponent; omega) hq) | |
| _ _ _ _ _ _ (permutation_bounded w n n hp) | |
| (fun _ => primeInitialized_bounded w n (compressionPrimeExponent n) | |
| (by unfold compressionPrimeExponent; omega) hq) | |
| intro stream cursor | |
| have h := hh stream cursor | |
| dsimp only [compression,compressionCore] | |
| change _+6 ≤ compressionWork n ∧ max 8 (_+8) ≤ (n+3)*(2*n^3+256) ∧ | |
| _+1 ≤ (n+2)*(n^3+64) | |
| refine ⟨?_,?_,?_⟩ | |
| · have hwork := Nat.add_le_add_right h.1 6 | |
| apply hwork.trans | |
| unfold compressionWork | |
| omega | |
| · apply max_le | |
| · nlinarith | |
| · have hc := Nat.add_le_add_right h.2.1 8 | |
| apply hc.trans | |
| have hA : (n+1)*(2*n^3+256)+8 ≤ (n+3)*(2*n^3+256) := by nlinarith | |
| have hB : ((n+1)*(n^3+64)+(2*n^3+124)+8)+8 ≤ | |
| (n+3)*(2*n^3+256) := by nlinarith | |
| omega | |
| · have hr := Nat.add_le_add_right h.2.2 1 | |
| exact hr.trans (by nlinarith) | |
| end LowSpaceSubsetSum.WordSeedMeasured | |
| end | |
| end OAI | |