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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 | import OAI.Combinatorics.EditApproximation.Geometry.ChargedPhysicalRequest
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalMedianQuery
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! A pass-zero seed read requests a cached coarse entry. Later seed reads
request the previous pass's final table value in each distinct copy.
Thus neither kind of seed value enters the charged program as a free oracle. -/
namespace EditApproximation
theorem chargedSeedQuery_value {M J ny passes copies T S : ℕ} (N : ℕ) (multiplier : ℕ → ℕ)
(parent : {r : PhysicalTableRequest M J ny // r.Bounded passes copies T S})
(q : TargetInterval ny) (answer : ChargedPhysicalRequest M J ny passes copies T S → ℚ) :
(chargedSeedQuery N multiplier parent (parent.val.node, q)).value (fun child => answer child.val) =
if parent.val.pass = 0 then answer (Sum.inl (parent.val.node, q)) else
(min N (upperMedian (List.ofFn fun copy : Fin copies =>
⌈(multiplier (parent.val.pass - 1) : ℚ) *
answer (Sum.inr (chargedPreviousPass parent q copy))⌉₊)) : ℕ) := by
unfold chargedSeedQuery
rw [dite_eq_left rfl]
split_ifs with hp
· rfl
· rw [FiniteQuery.collect_value]
simp only [chargedPreviousRead, FiniteQuery.value, List.map_map,
List.ofFn_eq_map, Function.comp_def]
theorem chargedSeedQuery_cost {M J ny passes copies T S : ℕ} (N : ℕ) (multiplier : ℕ → ℕ)
(parent : {r : PhysicalTableRequest M J ny // r.Bounded passes copies T S})
(state : PhysicalEntry M J ny)
(oracle : {child : ChargedPhysicalRequest M J ny passes copies T S //
child.rank < ChargedPhysicalRequest.rank (Sum.inr parent)} → ℚ) :
(chargedSeedQuery N multiplier parent state).cost oracle (fun _ => 1) =
if state.1 = parent.val.node then if parent.val.pass = 0 then 1 else copies else 0 := by
unfold chargedSeedQuery
split_ifs with hnode hpass
· rfl
· rw [FiniteQuery.cost_collect]
simp only [chargedPreviousRead, FiniteQuery.cost, Nat.add_zero, List.map_const',
List.length_finRange, List.sum_replicate, smul_eq_mul, Nat.mul_one]
· rfl
end EditApproximation
end OAI
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