openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldHistory /RecoveryBudget.lean
| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore | |
| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InverseLinearRecovery | |
| namespace OAI | |
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AllFieldHistoryRecovery | |
| open AllFieldHistory AllFieldHistorySupport | |
| attribute [local instance] Classical.propDecidable Classical.decEq | |
| variable {K tick : ℕ} | |
| def minimumDilation (allocation : Allocation) (ε : ℝ) : ℕ := | |
| Nat.ceil (3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6)) + 2 | |
| theorem minimumDilation_spec (allocation : Allocation) (ε : ℝ) {m : ℕ} | |
| (hm : minimumDilation (K := K) (tick := tick) allocation ε ≤ m) : | |
| 2 ≤ m ∧ | |
| 3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6) < | |
| (m : ℝ) := by | |
| constructor | |
| · exact (Nat.le_add_left 2 _).trans hm | |
| · have hc := Nat.le_ceil | |
| (3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6)) | |
| have hm' : (Nat.ceil | |
| (3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6)) : ℝ) + 2 ≤ m := by | |
| exact_mod_cast hm | |
| linarith | |
| def shiftCount (allocation : Allocation) (ε : ℝ) (m : ℕ) : ℕ := | |
| RecoveryGrowth.shiftCount (supportRate (K := K) (tick := tick) allocation) | |
| (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6) m | |
| theorem replication_subexponential (allocation : Allocation) (ε : ℝ) : | |
| Filter.Tendsto (fun m : ℕ => | |
| Real.log (Fintype.card (InverseLinearRecovery.MaskRectangles | |
| (shiftCount (K := K) (tick := tick) allocation ε m))) / m) | |
| Filter.atTop (nhds 0) := | |
| InverseLinearRecovery.subexponential_copies (supportRate_pos allocation) | |
| (by have := AllFieldHistoryMasks.lossConstant_nonneg (K := K) (tick := tick) allocation ε | |
| linarith) | |
| end MatrixMultiplication.AllFieldHistoryRecovery | |
| end | |
| end OAI | |