| import OAI.Combinatorics.EditApproximation.Queries.QueryRefinementFinish |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryPreparedKernelWork |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryContinuationReadCosts |
|
|
| namespace OAI |
|
|
| namespace EditApproximation.BinaryFraction |
|
|
| private theorem continuationBudget_mono (M P nr ng no nr' ng' no' B : ℕ) |
| (hr : nr ≤ nr') (hg : ng ≤ ng') (ho : no ≤ no') : |
| physicalContinuationBitBudget M P nr ng no B ≤ physicalContinuationBitBudget M P nr' ng' no' B := by |
| unfold physicalContinuationBitBudget wideActionsBitBudget localContinuationBitBudget bellmanBitBudget |
| dsimp only |
| gcongr |
|
|
| def queryRefinementFinishBitBudget (M P F B R G O : ℕ) : ℕ := |
| R + localRepresentativesBitBudget P F B * (R + 2) + |
| physicalContinuationBitBudget M P ((B + 1) * (64 * F * P + 2) ^ 2) G O (B + 4) * tableReadFactor R + 2 |
|
|
| theorem queryRefinementFinishWithWork_bounds {nx ny M : ℕ} (parent : TargetInterval nx) |
| (N P F B R G O : ℕ) (a delta : BinaryFraction) |
| (initial : TargetInterval ny → BinaryFraction × ℕ) (t : ℕ) |
| (earlierKeys : List (Fin M × ℕ × TargetInterval ny)) (earlierWords : List BinaryFraction) |
| (groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M)) |
| (currentKeys : List (Fin M × TargetInterval ny)) (currentWords : List BinaryFraction) |
| (q : TargetInterval ny) (hB : 10 ≤ B) (hP : 0 < P) (hFpos : 1 ≤ F) |
| (hN : Nat.size N ≤ B) (hnx : Nat.size nx ≤ B) (hny : Nat.size ny + 2 ≤ B) |
| (hM : Nat.size M ≤ B) (hPb : Nat.size P ≤ B) (hF : Nat.size F + 2 ≤ B) |
| (ha : a.width ≤ B) (hd : delta.width ≤ B) |
| (hi : ∀ r, (initial r).1.width ≤ B) (hR : ∀ r, (initial r).2 ≤ R) |
| (he : ∀ input, (queryHistoryReadWithWork earlierKeys earlierWords (t - 1) input).1.width ≤ B) |
| (heR : ∀ input, (queryHistoryReadWithWork earlierKeys earlierWords (t - 1) input).2 ≤ R) |
| (hc : ∀ i r, (queryChildReadWithWork currentKeys currentWords i r).1.width ≤ B) |
| (hcR : ∀ i r, (queryChildReadWithWork currentKeys currentWords i r).2 ≤ R) |
| (hgroups : ∀ action ∈ groups, action.connection.width ≤ connectionWordBudget M (B + 4)) |
| (honline : ∀ action ∈ online, action.connection.width ≤ connectionWordBudget M (B + 4)) |
| (hglen : groups.length ≤ G) (holen : online.length ≤ O) : |
| (queryRefinementFinishWithWork parent N P F a delta initial t earlierKeys earlierWords groups online |
| currentKeys currentWords q).1.width ≤ preparedKernelWordBudget M B ∧ |
| (queryRefinementFinishWithWork parent N P F a delta initial t earlierKeys earlierWords groups online |
| currentKeys currentWords q).2 ≤ queryRefinementFinishBitBudget M P F B R G O := by |
| let reps := localRepresentativeReadWithWork N P F a initial q |
| let old := queryHistoryReadWithWork earlierKeys earlierWords (t - 1) |
| let current := fun input : Fin M × TargetInterval ny => queryChildReadWithWork currentKeys currentWords input.1 input.2 |
| have hr := localRepresentativeReadWithWork_eq_cost N P F a initial q R hR |
| have hrCost := localRepresentativesWithWork_cost N P F B a (fun r => (initial r).1) q |
| (by omega) hN hF hFpos hPb hP (by omega) ha hi |
| have hrlen : reps.1.length ≤ (B + 1) * (64 * F * P + 2) ^ 2 := by |
| rw [show reps.1 = (localRepresentativesWithWork N P F a (fun r => (initial r).1) q).1 from hr.1] |
| exact localRepresentativesWithWork_length_le N P F B hP hFpos a (fun r => (initial r).1) q hN |
| have hrwidth : ∀ br ∈ reps.1, Nat.size br.1 ≤ B + 4 := by |
| intro br hbr |
| rw [show reps.1 = (localRepresentativesWithWork N P F a (fun r => (initial r).1) q).1 from hr.1] at hbr |
| exact (localRepresentativesWithWork_scale_bits N P F B hP a (fun r => (initial r).1) q hN br hbr).trans (by omega) |
| have htail := physicalContinuationWithWork_bounds parent reps.1 P F (B + 4) delta (initial q).1 |
| (fun input => (old input).1) (fun input => (current input).1) groups online q hP (by omega) |
| (by omega) (by omega) (by omega) (by omega) (by omega) hrwidth (hd.trans (by omega)) |
| ((hi q).trans (by omega)) (fun input => (he input).trans (by omega)) |
| (fun input => (hc input.1 input.2).trans (by omega)) hgroups honline |
| have htailValue := (physicalContinuationReadWithWork_eq_cost parent reps.1 P F delta (initial q).1 |
| old current groups online q R heR (fun input => hcR input.1 input.2)).1 |
| have htailRead := physicalContinuationReadWithWork_factor parent reps.1 P F delta (initial q).1 |
| old current groups online q R heR (fun input => hcR input.1 input.2) |
| have htailCost := htail.2.trans (continuationBudget_mono M P _ _ _ _ _ _ (B + 4) hrlen hglen holen) |
| have hrTotal : reps.2 ≤ localRepresentativesBitBudget P F B * (R + 2) := |
| hr.2.trans (Nat.mul_le_mul_right _ hrCost) |
| have htTotal := htailRead.trans (Nat.mul_le_mul_right (tableReadFactor R) htailCost) |
| constructor |
| · change (physicalContinuationReadWithWork parent reps.1 P F delta (initial q).1 old current groups online q).1.width ≤ _ |
| rw [htailValue] |
| exact htail.1 |
| · change (initial q).2 + reps.2 + |
| (physicalContinuationReadWithWork parent reps.1 P F delta (initial q).1 old current groups online q).2 + 2 ≤ _ |
| have hq := hR q |
| unfold queryRefinementFinishBitBudget |
| omega |
|
|
| end EditApproximation.BinaryFraction |
|
|
| end OAI |
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|
