openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /BackgroundBlockCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredBackgroundBlocks | |
| namespace OAI | |
| /-! # Copying the eight background rows from their shared array | |
| Each read uses the computed prefix-sum offset. Rows are retained once in an | |
| explicit vector, and the returned view indexes that vector. | |
| -/ | |
| namespace LowSpaceSubsetSum.BackgroundBlockCells | |
| open scoped BigOperators | |
| variable {n m : ℕ} | |
| def index (counts : Vector ℕ 8) (hc : (∑ i : Fin 8, counts[i.val]) = m) | |
| (i : Fin 8) (k : Fin counts[i.val]) : Fin m := | |
| ⟨(∑ j : Fin i.val, counts[(Fin.castLE i.isLt.le j).val])+k.val, by | |
| have h := (StoredBackgroundBlocks.index counts hc i k).isLt | |
| rwa [StoredBackgroundBlocks.index_value] at h⟩ | |
| theorem index_value (counts : Vector ℕ 8) (hc : (∑ i : Fin 8, counts[i.val]) = m) | |
| (i : Fin 8) (k : Fin counts[i.val]) : | |
| index counts hc i k = StoredBackgroundBlocks.index counts hc i k := by | |
| apply Fin.ext | |
| exact (StoredBackgroundBlocks.index_value counts hc i k).symm | |
| def row (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) (i : Fin 8) : | |
| (List (Fin n) × ℕ) × ℕ := | |
| let out := (List.finRange counts[i.val]).map (fun k => xs[(index counts hc i k).val]) | |
| ((out, 24*counts[i.val]+16), 6*counts[i.val]+32) | |
| theorem row_value (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) (i : Fin 8) : | |
| (row xs counts hc i).1.1 = StoredBackgroundBlocks.row xs counts hc i := by | |
| simp only [row, StoredBackgroundBlocks.row, List.ofFn_eq_map, index_value] | |
| def run (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) : | |
| (Vector (List (Fin n)) 8 × ℕ) × ℕ := | |
| let rows := Vector.ofFn (fun i => row xs counts hc i) | |
| ((rows.map (fun r => r.1.1), 32*m+200), | |
| (∑ i : Fin 8, rows[i.val].2)+4*m+128) | |
| theorem value (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) : | |
| (run xs counts hc).1 = StoredBackgroundBlocks.run xs counts hc := by | |
| apply Prod.ext | |
| · apply Vector.ext | |
| intro i hi | |
| simp only [run, StoredBackgroundBlocks.run, Vector.getElem_map, Vector.getElem_ofFn, row_value] | |
| · rfl | |
| theorem row_work (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) : | |
| (∑ i : Fin 8, (row xs counts hc i).1.2)+72 ≤ (run xs counts hc).1.2 := by | |
| simp only [row, run, Finset.sum_add_distrib, ← Finset.mul_sum, | |
| Finset.sum_const, Finset.card_univ, Fintype.card_fin, smul_eq_mul, hc] | |
| omega | |
| theorem cells (xs : Vector (Fin n) m) (counts : Vector ℕ 8) | |
| (hc : (∑ i : Fin 8, counts[i.val]) = m) : | |
| (run xs counts hc).2 ≤ 16*m+512 := by | |
| simp only [run, Vector.getElem_ofFn, row, Finset.sum_add_distrib, | |
| ← Finset.mul_sum, Finset.sum_const, Finset.card_univ, Fintype.card_fin, | |
| smul_eq_mul, hc] | |
| omega | |
| end LowSpaceSubsetSum.BackgroundBlockCells | |
| end OAI | |