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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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.Combinatorics.EditApproximation.Queries.QueryActiveCost
namespace OAI
namespace EditApproximation
section Bounded
variable {α : Type*} {M J R : ℕ} [NeZero M] [NeZero R]
(source target : List α) (N P F exponent H Q eM eF B E Scale : ℕ)
(hP : 0 < P) (hM : M = 2 ^ eM) (hF : F = 2 ^ eF)
(tau a eta kappa delta : BinaryFraction) (T S passes copies : ℕ)
(parent : {r : PhysicalTableRequest M J target.length // r.Bounded (passes + 1) copies T S})
(hindex : T < parent.val.index) (hbelow : parent.val.node.1.val < J)
(oracle : QueryBoundedChild (σ := PhysicalFamilyScalarKey (M := M) (J := J) (R := R)
target N P F exponent H Q passes copies) (passes + 1) copies T S parent → BinaryFraction)
(readWork : QueryBoundedChild (σ := PhysicalFamilyScalarKey (M := M) (J := J) (R := R)
target N P F exponent H Q passes copies) (passes + 1) copies T S parent → ℕ)
local notation "rawOracle" => queryRestrictedPhysicalOracle (passes + 1) copies T S parent oracle
local notation "rawWork" => queryRestrictedPhysicalReadWork (passes + 1) copies T S parent readWork
local notation "tableOracle" => (fun key => queryActiveOracle T parent.val hindex S rawOracle (Sum.inr key))
open BinaryFraction
theorem queryFamilyBoundedActive_work_bound (V C : ℕ)
(hn : PreparedKernelNumericBounds (nx := source.length) (ny := target.length) (M := M)
N P F exponent H Q eM eF B E Scale (parent.val.index - T - 1) tau a eta kappa delta (fun _ _ => 0))
(htime : Nat.size (parent.val.index - T) ≤ B) (hPd : ∃ j : ℕ, P = 2 ^ j)
(hV : 2 ≤ V) (hVB : V ≤ B) (hMean : warmupAverageWordBudget T V ≤ B)
(hc : ∀ key, readWork key ≤ C)
(hv : ∀ key, (oracle (Sum.inr key)).width ≤ V)
(hsv : ∀ key, (oracle (Sum.inl key)).width ≤ B)
(hnonneg : ∀ key, 0 ≤ (oracle (Sum.inr key)).value)
(hprior : ∀ input : Fin M × ℕ × TargetInterval target.length,
((queryPhysicalEarlierRead copies parent.val.pass parent.val.copy T S (parent.val.index - T)
parent.val.node hbelow parent.val.state input).value tableOracle).value ≤
((queryPhysicalWarmupRead copies parent.val.pass parent.val.copy T S (parent.val.index - T)
(Nat.sub_pos_of_lt hindex) parent.val.node parent.val.state
(physicalChild parent.val.node hbelow input.1) (by change parent.val.node.1.val ≤ parent.val.node.1.val + 1; omega)
0 (Nat.zero_le T) input.2.2).value tableOracle).value)
(hp : Nat.size parent.val.pass ≤ B) (hcopy : Nat.size parent.val.copy ≤ B)
(hps : Nat.size (passes + 1) ≤ B) (hcs : Nat.size (copies + 1) ≤ B)
(hT : Nat.size T ≤ B) (hS : Nat.size S ≤ B)
(hrank : Nat.size (parent.val.combinedRank T S) ≤ B)
(hbound : Nat.size (localOnlineMassBound M P H) ≤ B) :
(queryFamilyBoundedActive (R := R) source target N P F exponent H Q eM eF hP hM hF
tau a eta kappa delta T S passes copies parent hindex hbelow).work oracle readWork ≤
queryLocalProgramBitBudget M N P F exponent eM eF B Scale (parent.val.index - T)
(queryPhysicalDataBudget T J B V (50 * B + 21 + C + 1)
(queryFamilySourceReadBudget J B (50 * B + 21 + C + 1 + 1))) := by
rw [queryFamilyBoundedActive, queryRestrictPhysicalBody_work]
apply queryFamilyActive_work_bound source target N P F exponent H Q eM eF B E Scale hP hM hF
tau a eta kappa delta T S passes copies parent.val hindex hbelow rawOracle rawWork V
(50 * B + 21 + C) hn htime hPd hV hVB hMean
· intro key _ hcopy'
apply queryRestrictedPhysicalReadWork_bound (passes + 1) copies T S parent readWork B C
hps ((Nat.size_le_size (Nat.le_succ copies)).trans hcs) hT hS hrank hcopy hc (Sum.inr key)
intro child heq
cases heq
exact hcopy'
· intro key
apply queryRestrictedPhysicalReadWork_bound (passes + 1) copies T S parent readWork B C
hps ((Nat.size_le_size (Nat.le_succ copies)).trans hcs) hT hS hrank hcopy hc (Sum.inl key)
intro child heq
cases heq
· exact queryRestrictedPhysicalOracle_table_width (passes + 1) copies T S parent oracle V hV hv
· intro key
exact hsv key
· exact queryRestrictedPhysicalOracle_table_nonneg (passes + 1) copies T S parent oracle hnonneg
· exact hprior
· exact hp
· exact hcopy
· exact hps
· exact hcs
· exact hbound
theorem queryFamilyBoundedActive_width
(hn : PreparedKernelNumericBounds (nx := source.length) (ny := target.length) (M := M)
N P F exponent H Q eM eF B E Scale (parent.val.index - T - 1) tau a eta kappa delta (fun _ _ => 0))
(htime : Nat.size (parent.val.index - T) ≤ B)
(hv : ∀ key, (oracle (Sum.inr key)).width ≤ B)
(hnonneg : ∀ key, 0 ≤ (oracle (Sum.inr key)).value) :
((queryFamilyBoundedActive (R := R) source target N P F exponent H Q eM eF hP hM hF
tau a eta kappa delta T S passes copies parent hindex hbelow).value oracle).width ≤
preparedKernelWordBudget M B := by
rw [queryFamilyBoundedActive, queryRestrictPhysicalBody_value]
apply queryFamilyActive_width source target N P F exponent H Q eM eF B E Scale hP hM hF
tau a eta kappa delta T S passes copies parent.val hindex hbelow rawOracle hn htime
· exact queryRestrictedPhysicalOracle_table_width (passes + 1) copies T S parent oracle B
(by have := hn.largeBits; omega) hv
· exact queryRestrictedPhysicalOracle_table_nonneg (passes + 1) copies T S parent oracle hnonneg
end Bounded
end EditApproximation
end OAI
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