The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

0(x1) → 1(x1)
0(0(x1)) → 0(x1)
3(4(5(x1))) → 4(3(5(x1)))
2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 1(0(0(1(1(1(0(0(1(0(1(0(0(1(0(1(1(0(0(1(0(1(0(0(0(0(1(1(1(0(1(1(1(1(1(0(1(1(1(0(1(1(0(0(1(1(1(0(1(0(0(0(0(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(0(1(1(0(0(0(0(1(1(0(1(0(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(1(1(1(0(0(1(1(0(1(0(1(0(0(1(1(0(1(0(1(1(1(1(0(0(1(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(0(1(1(1(1(0(0(0(1(1(0(1(0(0(0(0(1(1(0(0(0(0(0(0(1(1(0(0(1(1(1(0(0(1(1(0(1(1(0(1(1(1(1(1(0(1(1(1(0(0(1(1(1(1(1(0(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
0(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

0(z0) → 1(z0)
0(0(z0)) → 0(z0)
0(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
3(4(5(z0))) → 4(3(5(z0)))
2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 1(0(0(1(1(1(0(0(1(0(1(0(0(1(0(1(1(0(0(1(0(1(0(0(0(0(1(1(1(0(1(1(1(1(1(0(1(1(1(0(1(1(0(0(1(1(1(0(1(0(0(0(0(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(0(1(1(0(0(0(0(1(1(0(1(0(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(1(1(1(0(0(1(1(0(1(0(1(0(0(1(1(0(1(0(1(1(1(1(0(0(1(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(0(1(1(1(1(0(0(0(1(1(0(1(0(0(0(0(1(1(0(0(0(0(0(0(1(1(0(0(1(1(1(0(0(1(1(0(1(1(0(1(1(1(1(1(0(1(1(1(0(0(1(1(1(1(1(0(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Tuples:

0'(0(z0)) → c1(0'(z0))
0'(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → c2(2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 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2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))), 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2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(z0))))))))))), 2'(2(2(2(2(2(2(2(2(2(z0)))))))))), 2'(2(2(2(2(2(2(2(2(z0))))))))), 2'(2(2(2(2(2(2(2(z0)))))))), 2'(2(2(2(2(2(2(z0))))))), 2'(2(2(2(2(2(z0)))))), 2'(2(2(2(2(z0))))), 2'(2(2(2(z0)))), 2'(2(2(z0))), 2'(2(z0)), 2'(z0))
3'(4(5(z0))) → c3(3'(5(z0)))
2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 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0'(0(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 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0'(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))), 0'(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))), 0'(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))), 0'(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))), 0'(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))), 0'(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))), 0'(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))))), 0'(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))), 0'(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))))))), 0'(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))), 0'(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))))))))))), 0'(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))), 0'(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))), 0'(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))), 0'(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))))), 0'(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))), 0'(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))))), 0'(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))), 0'(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))))), 0'(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))), 0'(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))), 0'(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))), 0'(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))), 0'(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))), 0'(0(1(1(1(1(1(1(1(1(1(z0))))))))))), 0'(1(1(1(1(1(1(1(1(1(z0)))))))))))

S tuples:

0'(0(z0)) → c1(0'(z0))
0'(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → c2(2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))), 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3'(4(5(z0))) → c3(3'(5(z0)))
2'(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 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0'(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))), 0'(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))), 0'(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))), 0'(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))), 0'(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))), 0'(0(1(1(1(1(1(1(1(1(1(z0))))))))))), 0'(1(1(1(1(1(1(1(1(1(z0)))))))))))

K tuples:none
Defined Rule Symbols:

0, 3, 2

Defined Pair Symbols:

0', 3', 2'

Compound Symbols:

c1, c2, c3, c4

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

0'(0(z0)) → c1(0'(z0))
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0'(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))), 0'(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))))))), 0'(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))), 0'(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))), 0'(0(0(1(1(1(1(1(1(1(1(1(z0)))))))))))), 0'(0(1(1(1(1(1(1(1(1(1(z0))))))))))), 0'(1(1(1(1(1(1(1(1(1(z0)))))))))))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

0(z0) → 1(z0)
0(0(z0)) → 0(z0)
0(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
3(4(5(z0))) → 4(3(5(z0)))
2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 1(0(0(1(1(1(0(0(1(0(1(0(0(1(0(1(1(0(0(1(0(1(0(0(0(0(1(1(1(0(1(1(1(1(1(0(1(1(1(0(1(1(0(0(1(1(1(0(1(0(0(0(0(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(0(1(1(0(0(0(0(1(1(0(1(0(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(1(1(1(0(0(1(1(0(1(0(1(0(0(1(1(0(1(0(1(1(1(1(0(0(1(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(0(1(1(1(1(0(0(0(1(1(0(1(0(0(0(0(1(1(0(0(0(0(0(0(1(1(0(0(1(1(1(0(0(1(1(0(1(1(0(1(1(1(1(1(0(1(1(1(0(0(1(1(1(1(1(0(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Tuples:

3'(4(5(z0))) → c3(3'(5(z0)))

S tuples:

3'(4(5(z0))) → c3(3'(5(z0)))

K tuples:none
Defined Rule Symbols:

0, 3, 2

Defined Pair Symbols:

3'

Compound Symbols:

c3

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

3'(4(5(z0))) → c3(3'(5(z0)))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

0(z0) → 1(z0)
0(0(z0)) → 0(z0)
0(1(1(1(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(1(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(0(1(0(1(0(0(0(0(0(1(1(1(1(0(0(1(0(0(0(0(1(1(1(1(0(1(1(0(0(1(0(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(1(1(1(0(1(0(0(1(0(1(1(1(0(0(1(1(0(1(0(1(0(1(1(0(0(1(0(0(1(0(0(0(1(0(1(0(1(0(1(0(0(0(1(0(0(1(0(1(1(1(0(0(0(1(0(1(0(0(1(1(0(1(1(1(1(1(0(1(0(1(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(0(0(0(0(0(0(0(0(0(1(0(1(0(0(0(1(0(1(1(0(0(1(1(1(1(1(0(0(1(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(1(0(1(0(1(0(0(1(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(1(1(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
3(4(5(z0))) → 4(3(5(z0)))
2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) → 1(0(0(1(1(1(0(0(1(0(1(0(0(1(0(1(1(0(0(1(0(1(0(0(0(0(1(1(1(0(1(1(1(1(1(0(1(1(1(0(1(1(0(0(1(1(1(0(1(0(0(0(0(1(1(0(0(1(1(1(0(1(1(1(1(0(1(0(0(0(1(1(0(0(0(0(1(1(0(1(0(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(1(1(1(0(0(1(1(0(1(0(1(0(0(1(1(0(1(0(1(1(1(1(0(0(1(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(0(1(1(1(1(0(0(0(1(1(0(1(0(0(0(0(1(1(0(0(0(0(0(0(1(1(0(0(1(1(1(0(0(1(1(0(1(1(0(1(1(1(1(1(0(1(1(1(0(0(1(1(1(1(1(0(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(1(1(1(0(0(0(0(0(1(1(0(0(1(0(1(0(1(1(0(0(0(0(0(0(0(0(1(0(0(0(0(1(1(1(1(1(1(1(1(1(z0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

0, 3, 2

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))