openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryGroupInputCost.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryGroupLabelCost | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryRoundedStateCost | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryRoundedCounts | |
| namespace OAI | |
| namespace EditApproximation.BinaryFraction | |
| def queryGroupInputsLength (M exponent : ℕ) : ℕ := | |
| M * ((24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4)) | |
| def queryGroupInputsBitBudget (M exponent B R : ℕ) : ℕ := | |
| let L := (24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4) | |
| queryGroupLabelsBitBudget M B R + | |
| M * (queryRoundedStatesBitBudget (24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) exponent B + | |
| 5 * L + 4) + 1 | |
| theorem queryGroupInputsWithWork_bounds {nx ny M : ℕ} (parent : TargetInterval nx) | |
| (b exponent Q B R : ℕ) (cell : ℕ × ℕ) (h : BinaryFraction) | |
| (draw : Fin (groupSampleCount M Q b h.value) → Fin M × ℕ) | |
| (hB : 4 ≤ B) (hMpos : 0 < M) (hnx : Nat.size nx ≤ B) (hny : Nat.size ny ≤ B) | |
| (hM : Nat.size M + 2 ≤ B) (hQ : Nat.size Q + 2 ≤ B) (hb : Nat.size b + 2 ≤ B) | |
| (hc : Nat.size cell.1 ≤ B) (he : exponent + 1 ≤ B) (hh : h.width ≤ B) | |
| (hhv : 0 ≤ h.value) (hfloor : (b : ℚ) / M ≤ h.value) (hR : ∀ i, (draw i).2 ≤ R) : | |
| (queryGroupInputsWithWork (ny := ny) parent b exponent Q cell h draw).1.length ≤ | |
| queryGroupInputsLength M exponent ∧ | |
| (queryGroupInputsWithWork (ny := ny) parent b exponent Q cell h draw).2 ≤ | |
| queryGroupInputsBitBudget M exponent B R := by | |
| let labels := queryGroupLabelsWithWork M Q b h draw | |
| let L := (24 * 2 ^ exponent * M + 2) * (4 * 2 ^ exponent + 4) | |
| let C := queryRoundedStatesBitBudget (24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) exponent B | |
| let run := fun i : Fin M => | |
| let states := queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h | |
| let tagged := arithmeticMapWithWork (fun r => ((i, r), 1)) states.1 | |
| (tagged.1, states.2 + tagged.2 + 1) | |
| have hl := queryGroupLabelsWithWork_bounds M Q b B R h draw hM hQ hb hh hR | |
| have hcounts := groupRoundSpacing_endpoint_counts M b exponent h.value hMpos hhv hfloor | |
| have hs (i : Fin M) := queryGroupRoundedStatesWithWork_bounds (ny := ny) parent b exponent B | |
| (24 * 2 ^ exponent * M + 2) (4 * 2 ^ exponent + 4) cell i h hB hnx hny (by omega) | |
| (by omega) hc he hh hcounts.1 hcounts.2 | |
| have hrun (i : Fin M) : (run i).1.length ≤ L ∧ (run i).2 ≤ C + 4 * L + 1 := by | |
| have hm := (arithmeticMapWithWork_cost (fun r => ((i, r), 1)) | |
| (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).1 1 | |
| (fun _ _ => le_rfl)).trans (Nat.mul_le_mul_right _ (hs i).1) | |
| constructor | |
| · simpa only [run, arithmeticMapWithWork_value, List.length_map] using (hs i).1 | |
| · change (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).2 + | |
| (arithmeticMapWithWork (fun r => ((i, r), 1)) | |
| (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).1).2 + 1 ≤ _ | |
| have hsc : (queryGroupRoundedStatesWithWork (ny := ny) parent b exponent cell i h).2 ≤ C := (hs i).2 | |
| change _ ≤ L * 4 at hm | |
| rw [Nat.mul_comm L 4] at hm | |
| omega | |
| have hlen := (arithmeticFlatMapWithWork_length run labels.1 L | |
| (fun i _ => (hrun i).1)).trans (Nat.mul_le_mul_right _ hl.1) | |
| have hcost := (arithmeticFlatMapWithWork_cost run labels.1 (C + 4 * L + 1) L | |
| (fun i _ => (hrun i).2) (fun i _ => (hrun i).1)).trans (Nat.mul_le_mul_right _ hl.1) | |
| rw [show C + 4 * L + 1 + L + 3 = C + 5 * L + 4 by omega] at hcost | |
| have hinputCost : labels.2 + (arithmeticFlatMapWithWork run labels.1).2 + 1 ≤ | |
| queryGroupInputsBitBudget M exponent B R := by | |
| have hlc : labels.2 ≤ queryGroupLabelsBitBudget M B R := hl.2 | |
| simpa only [queryGroupInputsBitBudget, C, L] using | |
| Nat.add_le_add_right (Nat.add_le_add hlc hcost) 1 | |
| simpa only [queryGroupInputsWithWork, labels, run, queryGroupInputsLength, L] using | |
| And.intro hlen hinputCost | |
| end EditApproximation.BinaryFraction | |
| end OAI | |