| import Mathlib.Algebra.Order.Floor.Semiring |
| import OAI.Computability.BinPacking.Configurations.K4Soundness |
| import OAI.Computability.BinPacking.Configurations.ReductionFractional |
|
|
| namespace OAI |
|
|
| namespace BinPackingGap |
|
|
| theorem completeFourGraph_reduction_bothLP_eq (c : ℕ) : |
| individualLP (reductionInstance c (1 / 8) completeFourGraph 2) = |
| (reductionBinBound c (1 / 8) completeFourGraph 2 : ℝ) ∧ |
| typeLP (reductionInstance c (1 / 8) completeFourGraph 2) = |
| (reductionBinBound c (1 / 8) completeFourGraph 2 : ℝ) := |
| reduction_bothLP_eq c (1 / 8) completeFourGraph 2 (by decide) |
|
|
| theorem structural_additive_gap (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) := by |
| refine ⟨reductionInstance c (1 / 8) completeFourGraph 2, |
| reductionBinBound c (1 / 8) completeFourGraph 2, |
| reductionInstance_count c (1 / 8) completeFourGraph 2 (by decide), |
| reductionInstance_size_bounds c (1 / 8) completeFourGraph 2, |
| completeFourGraph_reduction_opt_gap c, ?_⟩ |
| exact completeFourGraph_reduction_bothLP_eq c |
|
|
| theorem exists_mirup_counterexample : |
| ∃ I : Instance, |
| Nat.ceil (individualLP I) + 1 < opt I ∧ |
| Nat.ceil (typeLP I) + 1 < opt I := by |
| obtain ⟨I, B, _, _, hgap, hindividual, htype⟩ := structural_additive_gap 1 |
| refine ⟨I, ?_, ?_⟩ |
| · simpa only [hindividual, Nat.ceil_natCast] using hgap |
| · simpa only [htype, Nat.ceil_natCast] using hgap |
|
|
| theorem individual_mirup_false : |
| ¬ ∀ I : Instance, opt I ≤ Nat.ceil (individualLP I) + 1 := by |
| intro h |
| obtain ⟨I, hi, _⟩ := exists_mirup_counterexample |
| exact (not_le_of_gt hi) (h I) |
|
|
| theorem type_mirup_false : |
| ¬ ∀ I : Instance, opt I ≤ Nat.ceil (typeLP I) + 1 := by |
| intro h |
| obtain ⟨I, _, hi⟩ := exists_mirup_counterexample |
| exact (not_le_of_gt hi) (h I) |
|
|
| end BinPackingGap |
|
|
| end OAI |
|
|
