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namespace OAI
/-! # One actual residue bucket: left retention and right prefix search -/
namespace LowSpaceSubsetSum
structure RepresentationBucketData (n g : ℕ) where
left : RepresentationParentData n
right : RepresentationParentData n
coordinates : TopCoordinates n g
topOrder : List (Fin n)
namespace RepresentationBucketData
variable {n g : ℕ} (D : RepresentationBucketData n g)
def target : ℤ := (D.right.input.target : ℤ) -
weight (fun i => (D.right.input.values i : ℤ)) D.right.signs
def check (families : Fin 4 → Array (Finset (Fin g)))
(left right : Array (Entry (Fin n))) : Bool :=
(D.coordinates.check families D.target left.toList right.toList).accepted
/-- The left array is retained while each right task regenerates its own
restricted leaves and streams. An empty or overflowing left task is skipped. -/
def run (B : ℕ) (families : Fin 4 → Array (Finset (Fin g))) : Bool :=
let left := (D.left.collect B).state.user
if left.keys.overflow then false
else if left.records.isEmpty then false
else D.right.searchRight B (D.check families left.records) D.topOrder MaskRestriction.empty
theorem sound (B : ℕ) (families : Fin 4 → Array (Finset (Fin g)))
(hweights : D.left.weights = D.right.weights)
(htop : ∀ j : Fin 2, RepresentationUniverse.topDomain
(if j = 0 then D.left.leaves else D.right.leaves) = D.coordinates.domain)
(htlL : Disjoint (RepresentationUniverse.topDomain D.left.leaves)
(RepresentationUniverse.mixerDomain D.left.leaves))
(htbL : Disjoint (RepresentationUniverse.topDomain D.left.leaves)
(RepresentationUniverse.backgroundDomain D.left.leaves))
(hlbL : Disjoint (RepresentationUniverse.mixerDomain D.left.leaves)
(RepresentationUniverse.backgroundDomain D.left.leaves))
(htlR : Disjoint (RepresentationUniverse.topDomain D.right.leaves)
(RepresentationUniverse.mixerDomain D.right.leaves))
(htbR : Disjoint (RepresentationUniverse.topDomain D.right.leaves)
(RepresentationUniverse.backgroundDomain D.right.leaves))
(hlbR : Disjoint (RepresentationUniverse.mixerDomain D.right.leaves)
(RepresentationUniverse.backgroundDomain D.right.leaves))
(hshared : RepresentationUniverse.parentDomain D.left.leaves ∩
RepresentationUniverse.parentDomain D.right.leaves ⊆ D.coordinates.domain)
(hsize : D.left.capacity + 1 + (D.right.capacity + 1) ≤ 2 ^ n)
(hyes : D.run B families = true) : Solvable D.right.input := by
have htL : RepresentationUniverse.topDomain D.left.leaves = D.coordinates.domain := by
simpa only [ite_true] using htop 0
have htR : RepresentationUniverse.topDomain D.right.leaves = D.coordinates.domain := by
simpa only [show (1 : Fin 2) ≠ 0 from by decide, ite_false] using htop 1
unfold run at hyes
dsimp only at hyes
split at hyes
· contradiction
split at hyes
· contradiction
apply D.right.searchRight_subset_sum_sound B D.coordinates families (D.left.entries B)
D.topOrder MaskRestriction.empty (RepresentationUniverse.parentDomain D.left.leaves)
htR htlR htbR hlbR hshared ?_ ?_ hyes
· intro e he
rw [← hweights, ← htL]
exact D.left.entry_represents B htlL htbL hlbL e he
· exact (Nat.add_le_add_right (D.left.storage B).1 _).trans hsize
end RepresentationBucketData
end LowSpaceSubsetSum
end OAI
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