File size: 5,248 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 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Combinatorics.EditApproximation.Queries.QueryChargedFamily
import OAI.Combinatorics.EditApproximation.Queries.QueryChargedWarmupCost
namespace OAI
namespace EditApproximation
open BinaryFraction
def queryChargedTableBitBudget (M P F Amax copies J symbolBits positionBits B C active : ℕ) : ℕ :=
80 * (B + 1) + queryChargedWarmupBitBudget M P F Amax copies J symbolBits positionBits B (C + 1) + active
section Table
variable {M J R passes copies T S : ℕ} [NeZero M] [NeZero R]
(source target : List ℕ) (N P F exponent H Q eM eF Amax B C active : ℕ)
(hP : 0 < P) (hM : M = 2 ^ eM) (hF : F = 2 ^ eF)
(A a : ℕ → BinaryFraction) (tau eta kappa delta : BinaryFraction) (multiplier : ℕ → ℕ)
(occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ)
(parent : {r : PhysicalTableRequest M J target.length // r.Bounded (passes + 1) copies T S})
(oracle : ChargedFamilyQueryKey (R := R) target N P F exponent H Q (.inr parent) → BinaryFraction)
(readWork : ChargedFamilyQueryKey (R := R) target N P F exponent H Q (.inr parent) → ℕ)
local notation "tableOracle" => (fun key => oracle (Sum.inr key))
local notation "tableWork" => (fun key => 1 + readWork (Sum.inr key))
theorem queryChargedFamilyTable_work_bound
(hB : 10 ≤ B) (hN : Nat.size N ≤ B) (hPb : Nat.size P ≤ B)
(hFb : Nat.size F + 2 ≤ B) (hFpos : 1 ≤ F)
(hs : Nat.size source.length + 2 ≤ B) (ht : Nat.size target.length + 2 ≤ B)
(ht' : Nat.size (target.length + 1) + 2 ≤ B)
(hinit : Nat.size ((32 * F + 1) * N) + 3 ≤ B)
(hMb : Nat.size M ≤ B) (hJ : Nat.size J ≤ B) (hi : Nat.size parent.val.index ≤ B)
(hT : Nat.size T ≤ B)
(hA : (A parent.val.pass).width ≤ B) (ha : (a parent.val.pass).width ≤ B)
(hAone : 1 ≤ (A parent.val.pass).value) (hAmax : (A parent.val.pass).value ≤ Amax)
(hp : Nat.size parent.val.pass ≤ B) (hm : Nat.size (multiplier (parent.val.pass - 1)) + 2 ≤ B)
(hv : ∀ key, (oracle (Sum.inr key)).width ≤ B) (hc : ∀ key, readWork key ≤ C)
(hactive : ∀ (hindex : T < parent.val.index) (hbelow : parent.val.node.1.val < J),
(queryFamilyBoundedActive (R := R) source target N P F exponent H Q eM eF hP hM hF
tau (a parent.val.pass) eta kappa delta T S passes copies parent hindex hbelow).work
(fun key => oracle (Sum.map id (fun child =>
⟨Sum.inr child.val, ChargedPhysicalRequest.table_rank_lt parent child.val child.property⟩) key))
(fun key => 1 + readWork (Sum.map id (fun child =>
⟨Sum.inr child.val, ChargedPhysicalRequest.table_rank_lt parent child.val child.property⟩) key)) ≤ active) :
(queryChargedFamilyTable (R := R) source target N P F exponent H Q eM eF hP hM hF A a tau eta kappa
delta multiplier occurrence symbolBits positionBits parent).work oracle readWork ≤
queryChargedTableBitBudget M P F Amax copies J symbolBits positionBits B C active := by
have hw := queryChargedWarmup_work_bound source target N P F Amax B (C + 1)
(A parent.val.pass) (a parent.val.pass) multiplier parent occurrence symbolBits positionBits
tableOracle tableWork hB hP hN hPb hFb hFpos hs ht ht' hinit hMb hJ hi hA ha hAone hAmax hp hm hv
(fun key => by have := hc (Sum.inr key); omega)
have hinitial := queryPhysicalInitial_work_bound source target N P F Amax B (C + 1)
(A parent.val.pass) (a parent.val.pass) multiplier parent occurrence symbolBits positionBits
tableOracle tableWork parent.val.state (by omega) hP hN hPb (by omega) (by omega) (by omega)
hA ha hAone hAmax hp hm hv (fun key => by have := hc (Sum.inr key); omega)
have hiw : (queryPhysicalInitial source target N P F (A parent.val.pass) (a parent.val.pass)
multiplier parent occurrence symbolBits positionBits parent.val.state).work tableOracle tableWork ≤
queryChargedWarmupBitBudget M P F Amax copies J symbolBits positionBits B (C + 1) := by
apply hinitial.trans
unfold queryChargedWarmupBitBudget
dsimp only
omega
have hindexcost := wordLEWithWork_cost parent.val.index.bits T.bits
rw [Nat.size_eq_bits_len, Nat.size_eq_bits_len] at hindexcost
have hdepth := binaryNaturalCompareWithWork_cost parent.val.node.1.val J
have hn := (Nat.size_le_size (Nat.le_of_lt_succ parent.val.node.1.isLt)).trans hJ
have htest := querySliceTest_cost source (physicalSourceInterval source.length parent.val.node) B (by omega)
unfold queryChargedFamilyTable
simp only [BitQuery.work]
split
· simp only [BitQuery.mapKeysWithWork_work]
unfold queryChargedTableBitBudget
omega
· split
· split
· simp only [BitQuery.work, BitQuery.mapKeysWithWork_work]
unfold queryChargedTableBitBudget
omega
· rename_i hindex hbelow hlong
simp only [BitQuery.work, BitQuery.mapKeysWithWork_work]
have h := hactive (Nat.lt_of_not_ge (fun h => hindex ((queryNaturalLEWithWork_value _ _).mpr h)))
((binaryNaturalCompareWithWork_lt _ _).mp hbelow)
unfold queryChargedTableBitBudget
omega
· simp only [BitQuery.work, BitQuery.mapKeysWithWork_work]
unfold queryChargedTableBitBudget
omega
end Table
end EditApproximation
end OAI
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