openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryCountedActionSpace.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryCountedOptimizerSpace | |
| import OAI.Combinatorics.EditApproximation.Queries.QueryCoordinateSpace | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryEnvelopeSpace | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| namespace EditApproximation.BinaryFraction | |
| /-- All counted optimizer allocations, including coordinate construction. -/ | |
| theorem countedPreparedActionAllocation_le_work {M n : ℕ} (mass : Vector BinaryFraction M) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (footprint : ℕ → Fin M × TargetInterval n → ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ) | |
| (h : ∀ s coordinate, footprint s coordinate ≤ (read s coordinate).2) : | |
| countedPreparedActionAllocation mass read footprint eta kappa band G j t ≤ | |
| 3 * (countedPreparedActionReadWithWork mass read eta kappa band G j t).2 := by | |
| have hc := coordinateVectorAllocation_le_work band | |
| have hk := countedPreparedActionKernelAllocation_le_work mass read footprint eta kappa band G j t h | |
| have hlocal : (coordinateVectorWithWork band).2 ≤ | |
| (countedPreparedActionReadWithWork mass read eta kappa band G j t).2 := by | |
| unfold countedPreparedActionReadWithWork | |
| dsimp only | |
| omega | |
| unfold countedPreparedActionAllocation | |
| omega | |
| theorem countedBandActionAllocation_le_work {M n : ℕ} (b : ℕ) | |
| (initial : Fin M → TargetInterval n → BinaryFraction × ℕ) | |
| (initialAllocation : Fin M → TargetInterval n → ℕ) | |
| (read : ℕ → Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (footprint : ℕ → Fin M × TargetInterval n → ℕ) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ) | |
| (hi : ∀ i state, initialAllocation i state ≤ (initial i state).2) | |
| (hr : ∀ s coordinate, footprint s coordinate ≤ (read s coordinate).2) : | |
| countedBandActionAllocation b initial initialAllocation read footprint eta kappa band G j t ≤ | |
| 3 * (countedBandActionReadWithWork b initial read eta kappa band G j t).2 := by | |
| have he := envelopeVectorReadAllocation_le_work b band initial initialAllocation (fun i action _ => hi i (action i)) | |
| have hc := countedPreparedActionAllocation_le_work (envelopeVectorReadWithWork b band initial).1 | |
| read footprint eta kappa band G j t hr | |
| unfold countedBandActionAllocation countedBandActionReadWithWork | |
| dsimp only | |
| omega | |
| /-- The actual gathered-history specialization of the labeled action sampler. -/ | |
| def queryCountedActionAllocation {M n : ℕ} (mass : Vector BinaryFraction M) | |
| (keys : List (Fin M × ℕ × TargetInterval n)) (values : List BinaryFraction) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ) : ℕ := | |
| countedPreparedActionAllocation mass (queryHistoryReadWithWork keys values) | |
| (queryHistoryReadAllocation keys values) eta kappa band G j t | |
| theorem queryCountedActionAllocation_le_work {M n : ℕ} (mass : Vector BinaryFraction M) | |
| (keys : List (Fin M × ℕ × TargetInterval n)) (values : List BinaryFraction) | |
| (eta kappa : BinaryFraction) (band : List (Fin M → TargetInterval n)) (G j t : ℕ) : | |
| queryCountedActionAllocation mass keys values eta kappa band G j t ≤ | |
| 3 * (countedPreparedActionReadWithWork mass (queryHistoryReadWithWork keys values) | |
| eta kappa band G j t).2 := | |
| countedPreparedActionAllocation_le_work mass _ _ eta kappa band G j t | |
| (fun _ _ => queryHistoryReadAllocation_le_work _ _ _ _) | |
| end EditApproximation.BinaryFraction | |
| end OAI | |