File size: 3,691 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 | import OAI.Combinatorics.EditApproximation.Queries.QueryInitialGather
import OAI.Combinatorics.EditApproximation.Queries.QueryRepresentatives
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! The band-data stage constructs its keys by counted Boolean routines,
then gathers mass and initial words in the reference order. -/
namespace EditApproximation
theorem queryBandGather_refines {ι β : Type*} {nx ny : ℕ} {relation : β → ℚ → Prop}
(M : ℕ) (parent : TargetInterval nx) (N P F exponent : ℕ) (hP : 0 < P)
(a : BinaryFraction) (initial : TargetInterval ny → BinaryFraction × ℕ) (q : TargetInterval ny)
(R : ℕ) (hR : ∀ r, (initial r).2 ≤ R)
(massReader initialReader : (Fin M × TargetInterval ny) → BitQuery ι BinaryFraction)
(expectedMass expectedInitial : (Fin M × TargetInterval ny) → FiniteQuery ι)
(next : List (Fin M × TargetInterval ny) → List BinaryFraction → List BinaryFraction →
BitQuery ι β)
(expectedNext : List (Fin M × TargetInterval ny) → List ℚ → List ℚ → FiniteQuery ι)
(hm : ∀ key, BitQuery.Refines (fun word value => word.value = value)
(massReader key) (expectedMass key))
(hi : ∀ key, BitQuery.Refines (fun word value => word.value = value)
(initialReader key) (expectedInitial key))
(hn : ∀ mass words,
let keys := refinementBandInputList (M := M) parent N P F exponent a.value
(fun r => (initial r).1.value) q
BitQuery.Refines relation (next keys mass words)
(expectedNext keys (mass.map BinaryFraction.value) (words.map BinaryFraction.value))) :
BitQuery.Refines relation
(queryBandGather M parent N P F exponent hP a initial q massReader initialReader next)
(let keys := refinementBandInputList (M := M) parent N P F exponent a.value
(fun r => (initial r).1.value) q
FiniteQuery.collect expectedMass keys fun mass =>
FiniteQuery.collect expectedInitial keys (expectedNext keys mass)) := by
change BitQuery.Refines relation (.charge _ _) _
apply BitQuery.Refines.charge
change BitQuery.Refines relation (BitQuery.collect massReader _ _) _
rw [BinaryFraction.queryBandInputsWithWork_value M parent N P F exponent hP a initial q R hR]
apply BitQuery.Refines.collect massReader expectedMass hm
intro mass
exact BitQuery.Refines.collect initialReader expectedInitial hi _ _ _ (hn mass)
theorem queryBandGather_work {ι β : Type*} {nx ny : ℕ} (M : ℕ) (parent : TargetInterval nx)
(N P F exponent : ℕ) (hP : 0 < P) (a : BinaryFraction)
(initial : TargetInterval ny → BinaryFraction × ℕ) (q : TargetInterval ny)
(massReader initialReader : (Fin M × TargetInterval ny) → BitQuery ι BinaryFraction)
(next : List (Fin M × TargetInterval ny) → List BinaryFraction → List BinaryFraction →
BitQuery ι β) (oracle : ι → BinaryFraction) (readWork : ι → ℕ) :
let keys := BinaryFraction.queryBandInputsWithWork M parent N P F exponent hP a initial q
(queryBandGather M parent N P F exponent hP a initial q massReader initialReader next).work
oracle readWork = keys.2 +
(keys.1.map fun key => (massReader key).work oracle readWork + 3).sum +
(keys.1.map fun key => (initialReader key).work oracle readWork + 3).sum +
(next keys.1 (keys.1.map fun key => (massReader key).value oracle)
(keys.1.map fun key => (initialReader key).value oracle)).work oracle readWork := by
simp only [queryBandGather, BitQuery.work_bind, BitQuery.compute,
BitQuery.work, BitQuery.value, BitQuery.work_collect, Nat.add_zero, Nat.add_assoc]
end EditApproximation
end OAI
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