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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 | import OAI.Computability.BinPacking.Results.Aggregate
namespace OAI
namespace BinPackingGap
theorem main_results :
(∀ c : ℕ, ∃ I : Instance, ∃ B : ℕ,
I.n = 5 * B ∧
(∀ i, (1 / 6 : ℚ) < I.size i ∧ I.size i < 1) ∧
B + c < opt I ∧ individualLP I = (B : ℝ) ∧ typeLP I = (B : ℝ)) ∧
(∀ c : Nat, PackingGapNPHard c) ∧
((∃ c : Nat, Nonempty (AbsoluteAdditiveAlgorithm c)) ↔ PEqualsNP) ∧
(PEqualsNP → ∃ algorithm : AbsoluteAdditiveAlgorithm 0,
(∀ (items : RawInstance) (valid : items.Valid), ∃ packing : RawPacking,
algorithm.run (BinaryEncoding.rawInstanceBits items) = some packing ∧
packing.Feasible items ∧ packing.bins = opt (items.toInstance valid)) ∧
algorithm.run (BinaryEncoding.rawInstanceBits []) = some ⟨0, []⟩ ∧
(∀ input : List Bool, BinaryEncoding.decodeRawInstance input = none →
algorithm.run input = none) ∧
(∀ items : RawInstance, ¬ items.Valid →
algorithm.run (BinaryEncoding.rawInstanceBits items) = none)) := by
refine ⟨structural_additive_gap, packingGapNPHard,
exists_absoluteAdditiveAlgorithm_iff_pEqualsNP, ?_⟩
intro h
exact ⟨absoluteAdditiveAlgorithm_of_pEqualsNP h,
absoluteAdditiveAlgorithm_of_pEqualsNP_optimal h,
absoluteAdditiveAlgorithm_of_pEqualsNP_empty h,
absoluteAdditiveAlgorithm_of_pEqualsNP_malformed h,
absoluteAdditiveAlgorithm_of_pEqualsNP_invalid h⟩
end BinPackingGap
end OAI
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