(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
0(1(2(x))) → 1(3(2(x)))
1(1(4(2(3(x))))) → 1(5(4(4(x))))
2(2(1(3(5(x))))) → 1(2(0(5(x))))
3(0(0(2(1(x))))) → 0(5(4(0(x))))
4(3(2(5(0(x))))) → 3(4(1(5(5(x)))))
0(0(0(3(2(5(x)))))) → 2(0(5(5(4(5(x))))))
2(2(1(1(3(2(x)))))) → 2(1(4(0(0(x)))))
3(3(1(1(2(2(x)))))) → 3(3(0(0(2(x)))))
0(2(5(1(0(4(2(2(x)))))))) → 0(0(3(5(1(5(4(x)))))))
4(0(5(3(5(1(3(5(x)))))))) → 4(0(0(1(3(0(1(5(x))))))))
4(4(4(3(5(1(4(0(x)))))))) → 3(0(2(2(2(2(3(2(x))))))))
0(0(5(3(2(2(5(0(3(x))))))))) → 2(5(5(4(2(2(5(0(3(x)))))))))
3(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 0(4(3(0(4(4(2(4(1(4(0(5(x))))))))))))
4(3(3(3(5(0(0(3(2(4(4(1(2(x))))))))))))) → 4(4(5(2(2(0(5(0(1(4(3(0(x))))))))))))
4(4(3(3(1(2(2(5(3(5(3(2(3(x))))))))))))) → 4(1(4(0(0(2(5(4(4(2(0(3(x))))))))))))
0(0(5(0(1(4(4(3(5(2(0(0(3(3(x)))))))))))))) → 2(1(2(1(2(0(4(0(2(2(4(3(3(5(4(x)))))))))))))))
3(0(1(5(5(1(0(4(0(0(2(1(0(3(x)))))))))))))) → 3(4(0(1(2(5(2(2(0(3(0(4(5(1(x))))))))))))))
5(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 2(4(5(0(2(0(3(0(2(5(3(1(3(3(x))))))))))))))
1(0(4(3(2(1(1(1(2(4(4(5(5(0(1(x))))))))))))))) → 1(0(2(4(5(5(0(1(1(4(4(5(0(1(x))))))))))))))
3(4(1(1(4(4(0(4(4(2(4(1(0(0(5(3(2(x))))))))))))))))) → 3(5(5(5(0(1(3(2(4(2(0(3(5(3(0(x)))))))))))))))
0(2(1(3(5(3(4(1(1(4(4(0(4(3(4(1(0(2(x)))))))))))))))))) → 1(5(2(2(0(3(2(3(4(2(0(1(1(1(3(2(1(x)))))))))))))))))
3(3(0(0(1(2(3(5(3(0(5(2(0(0(2(4(4(1(x)))))))))))))))))) → 2(2(4(1(4(4(2(5(2(2(5(1(4(2(5(2(0(4(1(x)))))))))))))))))))
3(3(0(5(2(3(1(3(0(0(3(1(5(2(2(1(2(2(x)))))))))))))))))) → 1(0(3(0(4(2(4(3(2(0(4(2(1(5(5(2(2(x)))))))))))))))))
4(5(4(0(1(1(5(5(4(5(3(2(1(3(2(4(4(2(x)))))))))))))))))) → 3(4(5(5(3(0(4(3(3(3(0(5(3(2(2(5(0(x)))))))))))))))))
0(0(4(1(2(3(3(5(5(2(0(3(1(2(2(4(0(1(5(x))))))))))))))))))) → 2(2(0(1(0(1(5(0(1(0(0(5(0(1(1(4(5(x)))))))))))))))))
0(5(0(3(2(3(2(3(0(1(5(5(5(3(4(0(0(2(2(x))))))))))))))))))) → 2(2(5(4(0(0(1(5(5(3(2(2(5(0(0(5(4(0(x))))))))))))))))))
4(0(3(4(1(3(2(0(0(0(2(1(0(1(1(3(1(5(1(x))))))))))))))))))) → 0(3(5(3(4(5(1(0(0(3(1(0(2(4(1(3(3(0(x))))))))))))))))))
4(4(5(4(5(4(3(1(2(2(0(2(5(4(2(0(4(1(2(x))))))))))))))))))) → 0(3(5(5(2(0(3(4(2(4(5(0(2(1(3(2(2(1(x))))))))))))))))))
3(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 3(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))))))
2(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 5(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))))))))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0(x1)) = 17 + x1
POL(1(x1)) = 13 + x1
POL(2(x1)) = 13 + x1
POL(3(x1)) = 13 + x1
POL(4(x1)) = 12 + x1
POL(5(x1)) = 17 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
0(1(2(x))) → 1(3(2(x)))
1(1(4(2(3(x))))) → 1(5(4(4(x))))
2(2(1(3(5(x))))) → 1(2(0(5(x))))
3(0(0(2(1(x))))) → 0(5(4(0(x))))
0(0(0(3(2(5(x)))))) → 2(0(5(5(4(5(x))))))
2(2(1(1(3(2(x)))))) → 2(1(4(0(0(x)))))
3(3(1(1(2(2(x)))))) → 3(3(0(0(2(x)))))
0(2(5(1(0(4(2(2(x)))))))) → 0(0(3(5(1(5(4(x)))))))
0(0(5(3(2(2(5(0(3(x))))))))) → 2(5(5(4(2(2(5(0(3(x)))))))))
4(3(3(3(5(0(0(3(2(4(4(1(2(x))))))))))))) → 4(4(5(2(2(0(5(0(1(4(3(0(x))))))))))))
4(4(3(3(1(2(2(5(3(5(3(2(3(x))))))))))))) → 4(1(4(0(0(2(5(4(4(2(0(3(x))))))))))))
0(0(5(0(1(4(4(3(5(2(0(0(3(3(x)))))))))))))) → 2(1(2(1(2(0(4(0(2(2(4(3(3(5(4(x)))))))))))))))
3(0(1(5(5(1(0(4(0(0(2(1(0(3(x)))))))))))))) → 3(4(0(1(2(5(2(2(0(3(0(4(5(1(x))))))))))))))
1(0(4(3(2(1(1(1(2(4(4(5(5(0(1(x))))))))))))))) → 1(0(2(4(5(5(0(1(1(4(4(5(0(1(x))))))))))))))
3(4(1(1(4(4(0(4(4(2(4(1(0(0(5(3(2(x))))))))))))))))) → 3(5(5(5(0(1(3(2(4(2(0(3(5(3(0(x)))))))))))))))
0(2(1(3(5(3(4(1(1(4(4(0(4(3(4(1(0(2(x)))))))))))))))))) → 1(5(2(2(0(3(2(3(4(2(0(1(1(1(3(2(1(x)))))))))))))))))
3(3(0(0(1(2(3(5(3(0(5(2(0(0(2(4(4(1(x)))))))))))))))))) → 2(2(4(1(4(4(2(5(2(2(5(1(4(2(5(2(0(4(1(x)))))))))))))))))))
3(3(0(5(2(3(1(3(0(0(3(1(5(2(2(1(2(2(x)))))))))))))))))) → 1(0(3(0(4(2(4(3(2(0(4(2(1(5(5(2(2(x)))))))))))))))))
4(5(4(0(1(1(5(5(4(5(3(2(1(3(2(4(4(2(x)))))))))))))))))) → 3(4(5(5(3(0(4(3(3(3(0(5(3(2(2(5(0(x)))))))))))))))))
0(0(4(1(2(3(3(5(5(2(0(3(1(2(2(4(0(1(5(x))))))))))))))))))) → 2(2(0(1(0(1(5(0(1(0(0(5(0(1(1(4(5(x)))))))))))))))))
0(5(0(3(2(3(2(3(0(1(5(5(5(3(4(0(0(2(2(x))))))))))))))))))) → 2(2(5(4(0(0(1(5(5(3(2(2(5(0(0(5(4(0(x))))))))))))))))))
4(0(3(4(1(3(2(0(0(0(2(1(0(1(1(3(1(5(1(x))))))))))))))))))) → 0(3(5(3(4(5(1(0(0(3(1(0(2(4(1(3(3(0(x))))))))))))))))))
4(4(5(4(5(4(3(1(2(2(0(2(5(4(2(0(4(1(2(x))))))))))))))))))) → 0(3(5(5(2(0(3(4(2(4(5(0(2(1(3(2(2(1(x))))))))))))))))))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
4(3(2(5(0(x))))) → 3(4(1(5(5(x)))))
4(0(5(3(5(1(3(5(x)))))))) → 4(0(0(1(3(0(1(5(x))))))))
4(4(4(3(5(1(4(0(x)))))))) → 3(0(2(2(2(2(3(2(x))))))))
3(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 0(4(3(0(4(4(2(4(1(4(0(5(x))))))))))))
5(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 2(4(5(0(2(0(3(0(2(5(3(1(3(3(x))))))))))))))
3(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 3(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))))))
2(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 5(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))))))))
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
41(3(2(5(0(x))))) → 31(4(1(5(5(x)))))
41(3(2(5(0(x))))) → 41(1(5(5(x))))
41(3(2(5(0(x))))) → 51(5(x))
41(3(2(5(0(x))))) → 51(x)
41(0(5(3(5(1(3(5(x)))))))) → 41(0(0(1(3(0(1(5(x))))))))
41(0(5(3(5(1(3(5(x)))))))) → 31(0(1(5(x))))
41(4(4(3(5(1(4(0(x)))))))) → 31(0(2(2(2(2(3(2(x))))))))
41(4(4(3(5(1(4(0(x)))))))) → 21(2(2(2(3(2(x))))))
41(4(4(3(5(1(4(0(x)))))))) → 21(2(2(3(2(x)))))
41(4(4(3(5(1(4(0(x)))))))) → 21(2(3(2(x))))
41(4(4(3(5(1(4(0(x)))))))) → 21(3(2(x)))
41(4(4(3(5(1(4(0(x)))))))) → 31(2(x))
41(4(4(3(5(1(4(0(x)))))))) → 21(x)
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 41(3(0(4(4(2(4(1(4(0(5(x)))))))))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 31(0(4(4(2(4(1(4(0(5(x))))))))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 41(4(2(4(1(4(0(5(x))))))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 41(2(4(1(4(0(5(x)))))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 21(4(1(4(0(5(x))))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 41(1(4(0(5(x)))))
31(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 41(0(5(x)))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 21(4(5(0(2(0(3(0(2(5(3(1(3(3(x))))))))))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 41(5(0(2(0(3(0(2(5(3(1(3(3(x)))))))))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 51(0(2(0(3(0(2(5(3(1(3(3(x))))))))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 21(0(3(0(2(5(3(1(3(3(x))))))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 31(0(2(5(3(1(3(3(x))))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 21(5(3(1(3(3(x))))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 51(3(1(3(3(x)))))
51(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 31(1(3(3(x))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 31(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 41(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x))))))))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 31(1(3(5(5(5(4(0(4(1(2(5(5(0(x))))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 31(5(5(5(4(0(4(1(2(5(5(0(x))))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(5(5(4(0(4(1(2(5(5(0(x)))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(5(4(0(4(1(2(5(5(0(x))))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(4(0(4(1(2(5(5(0(x)))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 41(0(4(1(2(5(5(0(x))))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 41(1(2(5(5(0(x))))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 21(5(5(0(x))))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(5(0(x)))
31(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 51(0(x))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 51(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 41(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x))))))))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 51(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 41(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x))))))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 31(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x))))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 41(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 31(0(3(3(1(1(4(4(2(1(2(0(1(0(x))))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 31(3(1(1(4(4(2(1(2(0(1(0(x))))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 31(1(1(4(4(2(1(2(0(1(0(x)))))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 41(4(2(1(2(0(1(0(x))))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 41(2(1(2(0(1(0(x)))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 21(1(2(0(1(0(x))))))
21(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 21(0(1(0(x))))
The TRS R consists of the following rules:
4(3(2(5(0(x))))) → 3(4(1(5(5(x)))))
4(0(5(3(5(1(3(5(x)))))))) → 4(0(0(1(3(0(1(5(x))))))))
4(4(4(3(5(1(4(0(x)))))))) → 3(0(2(2(2(2(3(2(x))))))))
3(2(2(0(0(0(0(3(1(0(5(x))))))))))) → 0(4(3(0(4(4(2(4(1(4(0(5(x))))))))))))
5(5(0(5(4(4(4(3(4(0(5(4(3(3(x)))))))))))))) → 2(4(5(0(2(0(3(0(2(5(3(1(3(3(x))))))))))))))
3(3(3(5(2(0(0(3(3(1(2(5(2(1(3(1(0(5(2(2(x)))))))))))))))))))) → 3(4(5(0(5(3(1(3(5(5(5(4(0(4(1(2(5(5(0(x)))))))))))))))))))
2(5(4(3(4(4(4(4(2(1(2(2(1(5(5(2(5(0(1(1(1(x))))))))))))))))))))) → 5(4(5(4(1(3(4(3(0(3(3(1(1(4(4(2(1(2(0(1(0(x)))))))))))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 55 less nodes.
(6) TRUE