openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Queries /QueryCallbackPolynomial.lean
| import OAI.Combinatorics.EditApproximation.Queries.QueryFamilyRefinementCost | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryPowerStorage | |
| import OAI.Combinatorics.EditApproximation.Probability.FiniteBitPolynomial | |
| namespace OAI | |
| namespace EditApproximation | |
| private abbrev GP := FiniteBitPolynomial (Fin 6) | |
| private def gp_storedSumWordBudget (n B : GP) : GP := | |
| (B + 1) + FiniteBitPolynomial.bitLength n + 2 * (B + 2) + 2 * (1 + n * ((B + 1) + (B + 1))) + 1 | |
| private def gp_storedSumBitBudget (n B : GP) : GP := | |
| 1 + n * (100000 * (gp_storedSumWordBudget n B + 1) ^ 2) | |
| private def gp_bandMassTotalStorageBudget (M B : GP) : GP := | |
| FiniteBitPolynomial.bitLength M + B + 2 * (1 + M * (B + 1)) + 2 | |
| private def gp_warmupAverageBitBudget (n B : GP) : GP := | |
| gp_storedSumBitBudget n B + 6 * n + | |
| 2100 * (gp_bandMassTotalStorageBudget n B + FiniteBitPolynomial.bitLength n + 4) ^ 2 + FiniteBitPolynomial.bitLength n + 2 | |
| private def gp_queryPhysicalDataBudget (T J B V C W : GP) : GP := | |
| W + (T + 1 + T * (7 * J + 30 + (C + 1) + 3) + gp_warmupAverageBitBudget T V) + | |
| (8 * B + 7 * J + 32 + (C + 1)) + 1 | |
| private def gp_queryFamilySourceReadBudget (J B W : GP) : GP := | |
| 200 * (1000 * (B + 1) + 1000 + 2) ^ 2 + 120 * (B + 1) ^ 2 + 10 * J + 100 + W | |
| private def gp_bitPowerBitBudget (n B : GP) : GP := | |
| 1 + n * (200 * ((n + 1) * (B + 1) + 1) ^ 2) | |
| private def gp_powerDenominatorBitBudget (n A B : GP) : GP := | |
| gp_bitPowerBitBudget n B + 5000 * (A + (n * B + 1) + 2) ^ 2 + 1 | |
| private def callbackPolynomial : GP := | |
| gp_queryPhysicalDataBudget (.input 0) (.input 1) (.input 2) (.input 3) (.input 4) | |
| (gp_queryFamilySourceReadBudget (.input 1) (.input 2) (.input 5)) | |
| def queryCallbackShellDegree : ℕ := callbackPolynomial.degree | |
| theorem queryCallbackShell_fixedPower (T J B V C W X : ℕ) (hX : 2 ≤ X) | |
| (hT : T ≤ X) (hJ : J ≤ X) (hB : B ≤ X) (hV : V ≤ X) (hC : C ≤ X) (hW : W ≤ X) : | |
| BinaryFraction.queryPhysicalDataBudget T J B V C (BinaryFraction.queryFamilySourceReadBudget J B W) ≤ | |
| X ^ queryCallbackShellDegree := by | |
| have h := callbackPolynomial.eval_le_pow ![T, J, B, V, C, W] X hX (by | |
| intro i | |
| fin_cases i | |
| · exact hT | |
| · exact hJ | |
| · exact hB | |
| · exact hV | |
| · exact hC | |
| · exact hW) | |
| exact h | |
| private def normalizationPolynomial : GP := | |
| gp_powerDenominatorBitBudget (.input 0) (.input 1) (.input 2) | |
| def queryNormalizationBitDegree : ℕ := normalizationPolynomial.degree | |
| theorem queryNormalization_fixedPower (n A B X : ℕ) (hX : 2 ≤ X) | |
| (hn : n ≤ X) (hA : A ≤ X) (hB : B ≤ X) : | |
| BinaryFraction.powerDenominatorBitBudget n A B ≤ X ^ queryNormalizationBitDegree := by | |
| have h := normalizationPolynomial.eval_le_pow ![n, A, B, 0, 0, 0] X hX (by | |
| intro i | |
| fin_cases i | |
| · exact hn | |
| · exact hA | |
| · exact hB | |
| · exact Nat.zero_le _ | |
| · exact Nat.zero_le _ | |
| · exact Nat.zero_le _) | |
| exact h | |
| end EditApproximation | |
| end OAI | |