openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationBucketProgram.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.ParentPrefixSearch | |
| 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 | |