openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Compression /CompressionSampleCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Compression.CompressionSampleArrays | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.PermutationSamplerCells | |
| namespace OAI | |
| /-! # Live storage for the bounded compression sample | |
| The accepted digits are materialized once. The prime candidate tuple is | |
| retained, but not searched, when the digit sampler fails. Ideal backup | |
| values remain proof data and do not select an execution branch. | |
| -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum.CompressionSampleCells | |
| variable {n : ℕ} | |
| def run (D : CompressionData n) (hn : 168000000 ≤ n) (s : (D.sampler hn).Seed) : | |
| (Option (Vector (Fin n) n × CompressionSetup.PrimeSpace n) × ℕ) × ℕ := | |
| let digits := PermutationSamplerCells.run n n s.1 | |
| match digits.1.value with | |
| | none => ((none, digits.1.reads+digits.1.tests+1), digits.2+3*n^3+128) | |
| | some d => | |
| let v := PermutationArray.build n d | |
| let q := SamplerVectorCells.primeSeed | |
| (show 4 ≤ compressionPrimeExponent n by unfold compressionPrimeExponent; omega) n s.2 | |
| ((q.1.value.map (fun q => (v.1, q)), | |
| digits.1.reads+digits.1.tests+v.2+q.1.reads+q.1.tests+3), | |
| digits.2+PermutationArray.peakCells n+q.2+4*n+64) | |
| theorem value (D : CompressionData n) (hn : 168000000 ≤ n) (s : (D.sampler hn).Seed) : | |
| (run D hn s).1 = CompressionSampleArrays.run D hn s := by | |
| simp only [run, CompressionSampleArrays.run, PermutationSamplerCells.run_value, | |
| SamplerVectorCells.primeSeed_value, CompressionSampleArrays.prime] | |
| split <;> simp_all only | |
| theorem cells (D : CompressionData n) (hn : 168000000 ≤ n) (s : (D.sampler hn).Seed) : | |
| (run D hn s).2 ≤ (n+1)*(3*n^3+256) := by | |
| have hd := PermutationSamplerCells.run_cells n n s.1 | |
| have hp := PermutationArray.peakCells_bound n | |
| have hq := SamplerVectorCells.primeSeed_cells | |
| (show 4 ≤ compressionPrimeExponent n by unfold compressionPrimeExponent; omega) n s.2 | |
| simp only [run] | |
| split <;> dsimp only <;> | |
| nlinarith only [hd, hp, hq, Nat.zero_le (n^3)] | |
| theorem cells_scale (D : CompressionData n) (hn : 168000000 ≤ n) | |
| (s : (D.sampler hn).Seed) : (run D hn s).2 ≤ inputScale D.input^5 := by | |
| have hu : n+1 ≤ inputScale D.input := by unfold inputScale; omega | |
| have hu0 : 10000 ≤ inputScale D.input := by omega | |
| have hc := Nat.pow_le_pow_left (show n ≤ inputScale D.input by omega) 3 | |
| have hu3 : inputScale D.input ≤ inputScale D.input^3 := by | |
| simpa only [pow_one] using pow_le_pow_right' (by omega : 1 ≤ inputScale D.input) | |
| (by decide : 1 ≤ 3) | |
| have ht : 3*n^3+256 ≤ 4*inputScale D.input^3 := by | |
| nlinarith only [hc, hu0, hu3] | |
| have hm := Nat.mul_le_mul hu ht | |
| have hm' : (n+1)*(3*n^3+256) ≤ 4*inputScale D.input^4 := by | |
| convert hm using 1; ring | |
| have hp : 4*inputScale D.input^4 ≤ inputScale D.input^5 := by | |
| calc | |
| _ ≤ inputScale D.input*inputScale D.input^4 := Nat.mul_le_mul_right _ (by omega) | |
| _ = _ := (pow_succ' _ _).symm | |
| exact (cells D hn s).trans (hm'.trans hp) | |
| end LowSpaceSubsetSum.CompressionSampleCells | |
| end | |
| end OAI | |