openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryCachedContinuation.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryPhysicalContinuation | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryPowerStorage | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| /-! Cache insertion compresses the actual continuation result by executed | |
| Boolean multiplication and division. Its cost is part of the return | |
| counter; no rational normalization operation is assigned unit cost. -/ | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| theorem finishCachedWithWork_value (raw : BinaryFraction × ℕ) (base : List Bool) | |
| (hbase : 0 < bitWordValue base) (remaining : ℕ) | |
| (hden : raw.1.value.den ∣ bitWordValue base ^ remaining) : | |
| (finishCachedWithWork raw base hbase remaining).1.value = raw.1.value := | |
| powerDenominatorWithWork_value raw.1 base hbase remaining hden | |
| theorem finishCachedWithWork_bounds (raw : BinaryFraction × ℕ) (base : List Bool) | |
| (hbase : 0 < bitWordValue base) (remaining A B C V : ℕ) | |
| (ha : raw.1.width ≤ A) (hb : base.length ≤ B) (hc : raw.2 ≤ C) | |
| (hden : raw.1.value.den ∣ bitWordValue base ^ remaining) (hv : |raw.1.value| ≤ V) : | |
| (finishCachedWithWork raw base hbase remaining).1.width ≤ | |
| Nat.size V + 2 * (remaining * B + 1) + 1 ∧ | |
| (finishCachedWithWork raw base hbase remaining).2 ≤ | |
| C + powerDenominatorBitBudget remaining A B + 1 := by | |
| refine ⟨powerDenominatorWithWork_width raw.1 base hbase remaining V B hb hden hv, ?_⟩ | |
| exact Nat.add_le_add_right (Nat.add_le_add hc | |
| (powerDenominatorWithWork_cost raw.1 base hbase remaining A B ha hb)) 1 | |
| def cachedPhysicalContinuationWithWork {nx ny M : ℕ} (parent : TargetInterval nx) | |
| (representatives : List (ℕ × TargetInterval ny)) (P F : ℕ) | |
| (delta fallback : BinaryFraction) | |
| (old current : Fin M × TargetInterval ny → BinaryFraction) | |
| (groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M)) | |
| (query : TargetInterval ny) (base : List Bool) (hbase : 0 < bitWordValue base) (remaining : ℕ) : | |
| BinaryFraction × ℕ := | |
| finishCachedWithWork | |
| (physicalContinuationWithWork parent representatives P F delta fallback old current groups online query) | |
| base hbase remaining | |
| theorem cachedPhysicalContinuationWithWork_value {nx ny M : ℕ} (parent : TargetInterval nx) | |
| (representatives : List (ℕ × TargetInterval ny)) (P F : ℕ) | |
| (delta fallback : BinaryFraction) | |
| (old current : Fin M × TargetInterval ny → BinaryFraction) | |
| (groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M)) | |
| (query : TargetInterval ny) (base : List Bool) (hbase : 0 < bitWordValue base) (remaining : ℕ) | |
| (hP : 0 < P) | |
| (hden : (rationalWideStep parent representatives P (fun _ => fallback.value) (128 * F) delta.value | |
| (fun s => (old s).value) (fun s => (current s).value) | |
| (groups.map BinaryBellmanAction.decode) (online.map BinaryBellmanAction.decode) query).den ∣ | |
| bitWordValue base ^ remaining) : | |
| (cachedPhysicalContinuationWithWork parent representatives P F delta fallback old current groups online | |
| query base hbase remaining).1.value = | |
| rationalWideStep parent representatives P (fun _ => fallback.value) (128 * F) delta.value | |
| (fun s => (old s).value) (fun s => (current s).value) | |
| (groups.map BinaryBellmanAction.decode) (online.map BinaryBellmanAction.decode) query := by | |
| apply (finishCachedWithWork_value _ base hbase remaining ?_).trans | |
| (physicalContinuationWithWork_value parent representatives P F delta fallback old current groups online query hP) | |
| rwa [physicalContinuationWithWork_value parent representatives P F delta fallback old current groups online query hP] | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |