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ea76aac | 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 | import OAI.Computability.LoopMatching.MachineMain
import OAI.Computability.LoopMatching.ExpandCount
import OAI.Computability.LoopMatching.SparseBoundary
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching
theorem actual_algorithm_correct (G : Input) :
0 < (MachineMain.A G).2 ∧
(MachineMain.A G).1 ≤ (MachineMain.A G).2 * count G ∧
(MachineMain.A G).2 * count G ≤ 2 ^ (18 * G.n) * (MachineMain.A G).1 ∧
((MachineMain.A G).1 = 0 ↔ count G = 0) := by
by_cases h : 2 * G.ordinary.records.length + G.loops.length < G.n
· have hz := count_eq_zero_of_sparse_bound G h
simp [MachineMain.A, h, hz]
· rw [MachineMain.A, ite_eq_right h]
exact ⟨Approximation.output_den_pos G,
Approximation.output_bounds_of_expanded_count G
(fun j _ => expanded_count G j)⟩
theorem fullEndpoint : FullEndpoint := by
obtain ⟨machine, hfinite, hlength⟩ := MachineMain.computational_endpoint
exact ⟨MachineMain.A, machine, hfinite, hlength, actual_algorithm_correct⟩
end LoopMatching
end OAI
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