openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionSamplerExecution.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationSamplerExecution | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionSamplerCoupling | |
| namespace OAI | |
| /-! # Executed bounded permutation and prime draws for compression -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.CompressionData | |
| variable {n : ℕ} (D : CompressionData n) | |
| def samplerExecution (hn : 168000000 ≤ n) : SamplerExecution.Execution (D.sampler hn) := | |
| SamplerExecution.bind (SamplerExecution.permutation n n) _ _ | |
| (fun _ => SamplerExecution.prime (by unfold compressionPrimeExponent; omega) n) | |
| theorem samplerExecution_bounded (hn : 168000000 ≤ n) : | |
| (D.samplerExecution hn).Bounded (n * n ^ 3 + n ^ 3) (SamplerExecution.primeTestBudget n) := by | |
| have hp := SamplerExecution.permutation_bounded n n | |
| have hq := SamplerExecution.prime_bounded | |
| (show 4 ≤ compressionPrimeExponent n by unfold compressionPrimeExponent; omega) n | |
| have hq' := hq.mono le_rfl (Nat.mul_le_mul_left (n ^ 3) | |
| (Nat.pow_le_pow_right (by decide) (show | |
| (compressionPrimeExponent n + 1) / 2 ≤ (n + 1) / 2 by | |
| unfold compressionPrimeExponent | |
| omega))) | |
| intro s | |
| have hb := SamplerExecution.bind_bounded (SamplerExecution.permutation n n) _ _ | |
| (fun _ => SamplerExecution.prime (by unfold compressionPrimeExponent; omega) n) | |
| hp (fun _ => hq') | |
| simpa only [samplerExecution, SamplerExecution.primeTestBudget, Nat.zero_add] using hb s | |
| theorem samplerExecution_operations (hn : 168000000 ≤ n) (s : (D.sampler hn).Seed) : | |
| ((D.samplerExecution hn).run s).reads + ((D.samplerExecution hn).run s).tests ≤ | |
| 3 * inputScale D.input ^ 4 * 2 ^ ((n + 1) / 2) := by | |
| have h := D.samplerExecution_bounded hn s | |
| have hu : n ≤ inputScale D.input := by unfold inputScale; omega | |
| have hu1 : 1 ≤ inputScale D.input := by unfold inputScale; omega | |
| have h3 := Nat.pow_le_pow_left hu 3 | |
| have h34 : inputScale D.input ^ 3 ≤ inputScale D.input ^ 4 := | |
| pow_le_pow_right' hu1 (by decide) | |
| have h4 : n * n ^ 3 ≤ inputScale D.input ^ 4 := by | |
| have hm := Nat.mul_le_mul hu h3 | |
| simpa only [← pow_succ'] using hm | |
| have hread : n * n ^ 3 + n ^ 3 ≤ 2 * inputScale D.input ^ 4 := by omega | |
| have hT : 1 ≤ 2 ^ ((n + 1) / 2) := Nat.one_le_two_pow | |
| have hreads : n * n ^ 3 + n ^ 3 ≤ 2 * inputScale D.input ^ 4 * 2 ^ ((n + 1) / 2) := | |
| hread.trans (Nat.le_mul_of_pos_right _ (by omega)) | |
| have htests := Nat.mul_le_mul_right (2 ^ ((n + 1) / 2)) (h3.trans h34) | |
| unfold SamplerExecution.primeTestBudget at h | |
| nlinarith only [h.1, h.2, hreads, htests] | |
| end LowSpaceSubsetSum.CompressionData | |
| end | |
| end OAI | |