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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 96 97 98 99 100 101 102 | import OAI.Combinatorics.EditApproximation.Queries.QueryPhysicalInitialCost
import OAI.Combinatorics.EditApproximation.Queries.QueryChargedWarmup
import OAI.Combinatorics.EditApproximation.Queries.QueryWarmupProgramCost
namespace OAI
namespace EditApproximation
def queryChargedWarmupBitBudget (M P F Amax copies J symbolBits positionBits B C : ℕ) : ℕ :=
let I := queryPhysicalInitialBitBudget P F Amax copies J symbolBits positionBits B C
80 * (B + 1) + I +
BinaryFraction.queryWarmupProgramBitBudget M P F B (I + 7 * J + 33 + C)
section Warmup
variable (source target : List ℕ) {M J passes copies T S : ℕ}
(N P F Amax B C : ℕ) (A a : BinaryFraction) (multiplier : ℕ → ℕ)
(parent : {r : PhysicalTableRequest M J target.length // r.Bounded passes copies T S})
(occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ)
(oracle : {child : ChargedPhysicalRequest M J target.length passes copies T S //
child.rank < ChargedPhysicalRequest.rank (.inr parent)} → BinaryFraction)
(readWork : {child : ChargedPhysicalRequest M J target.length passes copies T S //
child.rank < ChargedPhysicalRequest.rank (.inr parent)} → ℕ)
theorem queryChargedWarmup_work_bound
(hB : 10 ≤ B) (hP : 0 < P) (hN : Nat.size N ≤ B) (hPb : Nat.size P ≤ B)
(hF : 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)
(hM : Nat.size M ≤ B) (hJ : Nat.size J ≤ B) (hi : Nat.size parent.val.index ≤ B)
(hA : A.width ≤ B) (ha : a.width ≤ B) (hAone : 1 ≤ A.value) (hAmax : A.value ≤ Amax)
(hp : Nat.size parent.val.pass ≤ B) (hm : Nat.size (multiplier (parent.val.pass - 1)) + 2 ≤ B)
(hv : ∀ key, (oracle key).width ≤ B) (hc : ∀ key, readWork key ≤ C) :
(queryChargedWarmup source target N P F A a multiplier parent occurrence symbolBits positionBits).work
oracle readWork ≤ queryChargedWarmupBitBudget M P F Amax copies J symbolBits positionBits B C := by
let I := queryPhysicalInitialBitBudget P F Amax copies J symbolBits positionBits B C
let reader := queryPhysicalInitial source target N P F A a multiplier parent occurrence symbolBits positionBits
have hiR (q : TargetInterval target.length) : (reader q).work oracle readWork ≤ I :=
queryPhysicalInitial_work_bound source target N P F Amax B C A a multiplier parent occurrence
symbolBits positionBits oracle readWork q (by omega) hP hN hPb (by omega) (by omega) (by omega)
hA ha hAone hAmax hp hm hv hc
have hiW (q : TargetInterval target.length) : ((reader q).value oracle).width ≤ B :=
queryPhysicalInitial_width source target N P F B A a multiplier parent occurrence symbolBits positionBits
oracle q hP ht' hinit
have hzero := naturalEqualWithWork_cost parent.val.index 0 B hi (by simp)
have hn := (Nat.size_le_size (Nat.le_of_lt_succ parent.val.node.1.isLt)).trans hJ
have hdepth := binaryNaturalCompareWithWork_cost parent.val.node.1.val J
have hdepth' : (binaryNaturalCompareWithWork parent.val.node.1.val J).2 ≤ 8 * B + 10 := by omega
have htest := querySliceTest_cost source (physicalSourceInterval source.length parent.val.node) B (by omega)
unfold queryChargedWarmup
simp only [BitQuery.work]
split
· change (naturalEqualWithWork parent.val.index 0).2 + 1 + (reader parent.val.state).work oracle readWork ≤ _
have h := hiR parent.val.state
unfold queryChargedWarmupBitBudget
dsimp only
dsimp only [I] at h
omega
· rename_i hnonzero
have hindex : 0 < parent.val.index :=
Nat.pos_of_ne_zero (fun h => hnonzero ((naturalEqualWithWork_value _ _).mpr h))
split
· split
· have h := hiR parent.val.state
change (naturalEqualWithWork parent.val.index 0).2 + 1 +
((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 +
(_ + (reader parent.val.state).work oracle readWork)) ≤ _
unfold queryChargedWarmupBitBudget
dsimp only
dsimp only [I] at h
omega
· rename_i hbelow hlong
have hcR (input : Fin M × TargetInterval target.length) :
(queryChargedWarmupChild parent ((binaryNaturalCompareWithWork_lt _ _).mp hbelow) hindex input).work
oracle readWork ≤ I + 7 * J + 33 + C := by
exact (queryChargedRead_work_bound parent oracle readWork C hc _).trans (by omega)
have hw := queryWarmupProgram_work_bound (physicalSourceInterval source.length parent.val.node)
N P F B (I + 7 * J + 33 + C) hP a reader
(queryChargedWarmupChild parent ((binaryNaturalCompareWithWork_lt _ _).mp hbelow) hindex) parent.val.state
oracle readWork hB hN hF hFpos hPb hs ht hM ha hiW (fun input => hv _)
(fun r => (hiR r).trans (by omega)) hcR
change (naturalEqualWithWork parent.val.index 0).2 + 1 +
((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 +
(_ + (queryWarmupProgram _ N P F a reader _ parent.val.state).work oracle readWork)) ≤ _
unfold queryChargedWarmupBitBudget
dsimp only
dsimp only [I] at hw
omega
· have h := hiR parent.val.state
change (naturalEqualWithWork parent.val.index 0).2 + 1 +
((binaryNaturalCompareWithWork parent.val.node.1.val J).2 + 1 +
(reader parent.val.state).work oracle readWork) ≤ _
unfold queryChargedWarmupBitBudget
dsimp only
dsimp only [I] at h
omega
end Warmup
end EditApproximation
end OAI
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