File size: 3,272 Bytes
8df23f4 | 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 67 68 | 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
|