File size: 5,361 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | 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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