| import OAI.Computability.BinPacking.Search.SearchInitializeEncoding | |
| namespace OAI | |
| namespace BinPackingGap.CapacityScan | |
| def step (label : Nat) (accumulator : Nat × Nat) (row : (Nat × Nat) × Nat) : Nat × Nat := | |
| if row.2 = label then RawPacking.addFraction accumulator row.1 else accumulator | |
| def scan (label : Nat) : RawInstance → List Nat → (Nat × Nat) → Bool | |
| | [], _, accumulator => decide (accumulator.1 ≤ accumulator.2) | |
| | _, [], accumulator => decide (accumulator.1 ≤ accumulator.2) | |
| | item :: items, assigned :: assignments, accumulator => | |
| if accumulator.1 ≤ accumulator.2 then | |
| scan label items assignments (step label accumulator (item, assigned)) | |
| else false | |
| theorem scan_eq_true_iff (label : Nat) (items : RawInstance) | |
| (valid : items.Valid) (assignments : List Nat) (accumulator : Nat × Nat) : | |
| scan label items assignments accumulator = true ↔ | |
| ((items.zip assignments).foldl (step label) accumulator).1 ≤ | |
| ((items.zip assignments).foldl (step label) accumulator).2 := by | |
| induction items generalizing assignments accumulator with | |
| | nil => simp [scan] | |
| | cons item items ih => | |
| have validTail : RawInstance.Valid items := | |
| fun q mem => valid q (List.mem_cons_of_mem item mem) | |
| cases assignments with | |
| | nil => simp [scan] | |
| | cons assigned assignments => | |
| by_cases accepted : accumulator.1 ≤ accumulator.2 | |
| · simpa only [scan, ite_eq_left accepted, List.zip_cons_cons, List.foldl_cons] using | |
| ih validTail assignments (step label accumulator (item, assigned)) | |
| · have overflow := FractionWidth.fold_overflow | |
| (item :: items) (assigned :: assignments) label accumulator valid | |
| (Nat.lt_of_not_ge accepted) | |
| have final_rejected : | |
| ¬(( (item :: items).zip (assigned :: assignments)).foldl | |
| (step label) accumulator).1 ≤ | |
| (((item :: items).zip (assigned :: assignments)).foldl | |
| (step label) accumulator).2 := by | |
| change (((item :: items).zip (assigned :: assignments)).foldl | |
| (step label) accumulator).2 < | |
| (((item :: items).zip (assigned :: assignments)).foldl | |
| (step label) accumulator).1 at overflow | |
| exact Nat.not_le_of_lt overflow | |
| simp only [scan, ite_eq_right accepted, Bool.false_eq_true, final_rejected] | |
| theorem scan_initial_eq_capacityCheck (label : Nat) (items : RawInstance) | |
| (valid : items.Valid) (assignments : List Nat) : | |
| scan label items assignments (0, 1) = | |
| ExtensionCertificate.capacityCheck items assignments label := by | |
| apply Bool.eq_iff_iff.mpr | |
| rw [scan_eq_true_iff label items valid assignments, | |
| ExtensionCertificate.capacityCheck_eq_true_iff] | |
| rfl | |
| theorem scan_initial_eq_true_iff_load (label : Nat) (items : RawInstance) | |
| (valid : items.Valid) (assignments : List Nat) : | |
| scan label items assignments (0, 1) = true ↔ | |
| RawPacking.load items assignments label ≤ 1 := by | |
| rw [scan_initial_eq_capacityCheck label items valid assignments] | |
| exact ExtensionCertificate.capacityCheck_eq_true_iff_load items valid assignments label | |
| end BinPackingGap.CapacityScan | |
| end OAI | |