openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Storage /ConcretePrefixCells.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.BudgetedPrefixCells | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.BudgetedConcretePrefix | |
| namespace OAI | |
| /-! # The stored collector inside the bounded prefix-memory traversal -/ | |
| namespace LowSpaceSubsetSum.ConcretePrefixCells | |
| variable {n : ℕ} | |
| def run (D : RepresentationParentData n) (F : IndexedMask n) | |
| (hF : F.semantic = D.signs) (B : ℕ) | |
| (check : Array (StoredEntry n) → ℕ → BudgetedPrefixCells.Execution Bool) | |
| (order : List (Fin n)) (P : StoredRestriction n) (budget : ℕ) := | |
| BudgetedPrefixCells.run (BudgetedPrefixCells.collect D F hF B) check | |
| ParentLiveState.dictionary order P budget | |
| theorem value (D : RepresentationParentData n) (F : IndexedMask n) | |
| (hF : F.semantic = D.signs) (B : ℕ) | |
| (check : Array (StoredEntry n) → ℕ → BudgetedPrefixCells.Execution Bool) | |
| (order : List (Fin n)) (P : StoredRestriction n) (budget : ℕ) : | |
| (run D F hF B check order P budget).1 = | |
| BudgetedConcretePrefix.run D F hF B (fun a b => (check a b).1) order P budget := by | |
| rw [run, BudgetedPrefixCells.value] | |
| simp only [BudgetedPrefixCells.collect_value, BudgetedConcretePrefix.run] | |
| theorem dictionary_bound (I : Instance n) (C : ℕ) (hn : 10000 ≤ n) | |
| (d : StoredEntry.Dictionary I C) : | |
| ParentLiveState.dictionary d ≤ inputScale I ^ 2 * (C + 1) := by | |
| have hd := ParentLiveState.dictionary_scale I C d | |
| have hu : 10000 ≤ inputScale I := by unfold inputScale; omega | |
| have hc : 26 * inputScale I ≤ inputScale I ^ 2 := by nlinarith only [hu] | |
| have hm := Nat.mul_le_mul_right (C + 1) hc | |
| have hbase : inputScale I ≤ inputScale I * (C + 1) := Nat.le_mul_of_pos_right _ (by omega) | |
| nlinarith only [hd, hu, hm, hbase] | |
| theorem cells (D : RepresentationParentData n) (F : IndexedMask n) | |
| (hF : F.semantic = D.signs) (B : ℕ) | |
| (check : Array (StoredEntry n) → ℕ → BudgetedPrefixCells.Execution Bool) | |
| (K : ℕ) | |
| (hcheck : ∀ (d : StoredEntry.Dictionary D.input D.capacity) b, | |
| (check d.val.records b).2 ≤ K) | |
| (order : List (Fin n)) (P : StoredRestriction n) (budget : ℕ) | |
| (hn : 10000 ≤ n) (hl : ∀ j, (D.pair j).left.occurrences.card ≤ D.capacity) | |
| (hr : ∀ j, (D.pair j).right.occurrences.card ≤ D.capacity) | |
| (hC : D.capacity ≤ 2 ^ n) (hB : B ≤ D.capacity) : | |
| (run D F hF B check order P budget).2 ≤ | |
| BudgetedPrefixCells.frame n * (order.length + 1) + | |
| inputScale D.input ^ 14 * (D.capacity + 1) + | |
| inputScale D.input ^ 2 * (D.capacity + 1) + K := | |
| BudgetedPrefixCells.cells _ _ _ _ _ _ | |
| (fun p b => BudgetedPrefixCells.collect_cells D F hF B p b hn hl hr hC hB) | |
| (dictionary_bound D.input D.capacity hn) hcheck order P budget | |
| end LowSpaceSubsetSum.ConcretePrefixCells | |
| end OAI | |