13.14/4.37 YES 13.25/4.41 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 13.25/4.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 13.25/4.41 13.25/4.41 13.25/4.41 Termination w.r.t. Q of the given QTRS could be proven: 13.25/4.41 13.25/4.41 (0) QTRS 13.25/4.41 (1) QTRSRRRProof [EQUIVALENT, 160 ms] 13.25/4.41 (2) QTRS 13.25/4.41 (3) DependencyPairsProof [EQUIVALENT, 32 ms] 13.25/4.41 (4) QDP 13.25/4.41 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 13.25/4.41 (6) TRUE 13.25/4.41 13.25/4.41 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (0) 13.25/4.41 Obligation: 13.25/4.41 Q restricted rewrite system: 13.25/4.41 The TRS R consists of the following rules: 13.25/4.41 13.25/4.41 0(1(1(1(x1)))) -> 2(1(3(x1))) 13.25/4.41 1(2(3(1(x1)))) -> 1(1(1(1(x1)))) 13.25/4.41 1(2(0(3(4(x1))))) -> 3(0(2(2(x1)))) 13.25/4.41 3(2(0(5(3(5(x1)))))) -> 3(1(2(4(2(5(x1)))))) 13.25/4.41 1(3(4(3(1(5(3(x1))))))) -> 5(2(0(0(4(0(3(x1))))))) 13.25/4.41 4(2(2(4(2(4(4(x1))))))) -> 4(2(2(3(0(4(x1)))))) 13.25/4.41 1(3(3(3(4(4(1(1(x1)))))))) -> 1(2(4(4(3(4(0(1(x1)))))))) 13.25/4.41 3(0(0(2(5(0(0(1(x1)))))))) -> 0(2(4(1(4(2(0(1(x1)))))))) 13.25/4.41 4(0(1(0(0(1(2(2(1(x1))))))))) -> 2(4(0(0(4(5(5(5(x1)))))))) 13.25/4.41 2(1(3(3(0(1(3(2(5(1(x1)))))))))) -> 2(2(0(1(2(1(2(1(1(3(x1)))))))))) 13.25/4.41 3(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 4(3(5(2(3(3(1(3(0(3(x1)))))))))) 13.25/4.41 3(4(3(5(1(3(1(2(2(5(x1)))))))))) -> 1(0(4(0(1(5(3(5(2(1(x1)))))))))) 13.25/4.41 5(0(1(1(0(2(2(0(1(1(x1)))))))))) -> 4(5(5(5(2(3(2(0(3(x1))))))))) 13.25/4.41 4(0(1(1(0(0(1(5(3(5(0(1(x1)))))))))))) -> 2(3(4(4(5(4(5(5(0(4(2(x1))))))))))) 13.25/4.41 5(5(4(2(0(5(2(4(3(2(5(5(x1)))))))))))) -> 3(4(0(0(1(1(3(5(1(0(1(4(x1)))))))))))) 13.25/4.41 3(4(0(1(1(5(3(5(0(4(4(3(4(3(x1)))))))))))))) -> 3(5(5(4(0(1(1(1(1(4(4(0(0(3(x1)))))))))))))) 13.25/4.41 4(0(0(2(4(5(4(2(1(4(3(4(4(3(x1)))))))))))))) -> 4(4(0(1(5(2(2(1(1(0(1(3(1(3(x1)))))))))))))) 13.25/4.41 1(5(2(4(4(0(2(4(5(5(1(2(3(3(1(x1))))))))))))))) -> 3(1(2(4(5(4(0(3(2(1(5(5(1(3(1(x1))))))))))))))) 13.25/4.41 2(0(2(0(5(5(5(1(4(4(4(3(3(3(0(x1))))))))))))))) -> 5(3(3(3(2(5(0(5(4(4(5(3(3(0(x1)))))))))))))) 13.25/4.41 5(0(4(2(1(1(2(4(1(0(3(1(0(3(5(x1))))))))))))))) -> 4(5(3(2(3(1(1(0(1(3(5(3(0(1(x1)))))))))))))) 13.25/4.41 5(0(4(5(4(0(4(0(4(5(2(0(1(2(1(x1))))))))))))))) -> 1(2(2(2(0(4(0(5(1(3(4(4(1(1(4(x1))))))))))))))) 13.25/4.41 5(3(1(2(3(2(2(2(4(1(0(4(3(3(4(0(x1)))))))))))))))) -> 5(5(4(0(2(5(0(0(1(3(0(3(4(3(4(0(x1)))))))))))))))) 13.25/4.41 1(3(4(1(3(3(0(2(3(3(3(2(5(4(4(4(4(x1))))))))))))))))) -> 1(1(3(2(5(0(4(2(4(3(4(0(3(3(3(0(4(x1))))))))))))))))) 13.25/4.41 3(4(3(3(4(3(2(4(1(1(2(1(4(5(3(0(0(x1))))))))))))))))) -> 1(1(3(0(5(3(2(0(4(4(1(3(5(0(0(2(0(x1))))))))))))))))) 13.25/4.41 5(3(3(4(5(3(5(1(2(5(5(0(5(5(1(5(3(x1))))))))))))))))) -> 5(5(3(5(2(1(3(3(5(1(3(5(0(5(4(3(x1)))))))))))))))) 13.25/4.41 2(0(1(2(0(3(5(1(4(5(2(3(5(2(4(1(0(3(x1)))))))))))))))))) -> 1(4(5(4(0(5(5(1(3(0(1(4(4(0(3(3(4(3(x1)))))))))))))))))) 13.25/4.41 5(1(1(0(2(2(1(1(5(5(2(2(4(0(4(3(2(2(x1)))))))))))))))))) -> 3(1(1(1(2(2(1(5(4(3(0(2(3(5(4(2(2(x1))))))))))))))))) 13.25/4.41 2(2(3(0(5(2(3(4(4(0(3(3(0(2(0(2(4(1(1(x1))))))))))))))))))) -> 2(1(5(2(1(1(5(4(5(4(3(3(5(4(5(5(3(4(x1)))))))))))))))))) 13.25/4.41 1(3(5(3(3(3(4(0(1(2(3(3(0(4(3(0(3(4(4(5(x1)))))))))))))))))))) -> 4(2(4(5(4(5(1(4(4(5(5(0(2(1(1(1(1(4(2(5(1(x1))))))))))))))))))))) 13.25/4.41 5(0(4(2(0(0(0(3(3(1(5(2(2(5(1(3(4(3(0(5(1(x1))))))))))))))))))))) -> 0(3(1(5(4(3(1(1(3(0(3(2(0(0(2(2(2(1(4(4(1(x1))))))))))))))))))))) 13.25/4.41 13.25/4.41 Q is empty. 13.25/4.41 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (1) QTRSRRRProof (EQUIVALENT) 13.25/4.41 Used ordering: 13.25/4.41 Polynomial interpretation [POLO]: 13.25/4.41 13.25/4.41 POL(0(x_1)) = 23 + x_1 13.25/4.41 POL(1(x_1)) = 21 + x_1 13.25/4.41 POL(2(x_1)) = 26 + x_1 13.25/4.41 POL(3(x_1)) = 27 + x_1 13.25/4.41 POL(4(x_1)) = 25 + x_1 13.25/4.41 POL(5(x_1)) = 25 + x_1 13.25/4.41 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 13.25/4.41 13.25/4.41 0(1(1(1(x1)))) -> 2(1(3(x1))) 13.25/4.41 1(2(3(1(x1)))) -> 1(1(1(1(x1)))) 13.25/4.41 1(2(0(3(4(x1))))) -> 3(0(2(2(x1)))) 13.25/4.41 3(2(0(5(3(5(x1)))))) -> 3(1(2(4(2(5(x1)))))) 13.25/4.41 1(3(4(3(1(5(3(x1))))))) -> 5(2(0(0(4(0(3(x1))))))) 13.25/4.41 4(2(2(4(2(4(4(x1))))))) -> 4(2(2(3(0(4(x1)))))) 13.25/4.41 1(3(3(3(4(4(1(1(x1)))))))) -> 1(2(4(4(3(4(0(1(x1)))))))) 13.25/4.41 3(0(0(2(5(0(0(1(x1)))))))) -> 0(2(4(1(4(2(0(1(x1)))))))) 13.25/4.41 4(0(1(0(0(1(2(2(1(x1))))))))) -> 2(4(0(0(4(5(5(5(x1)))))))) 13.25/4.41 2(1(3(3(0(1(3(2(5(1(x1)))))))))) -> 2(2(0(1(2(1(2(1(1(3(x1)))))))))) 13.25/4.41 3(4(3(5(1(3(1(2(2(5(x1)))))))))) -> 1(0(4(0(1(5(3(5(2(1(x1)))))))))) 13.25/4.41 5(0(1(1(0(2(2(0(1(1(x1)))))))))) -> 4(5(5(5(2(3(2(0(3(x1))))))))) 13.25/4.41 4(0(1(1(0(0(1(5(3(5(0(1(x1)))))))))))) -> 2(3(4(4(5(4(5(5(0(4(2(x1))))))))))) 13.25/4.41 5(5(4(2(0(5(2(4(3(2(5(5(x1)))))))))))) -> 3(4(0(0(1(1(3(5(1(0(1(4(x1)))))))))))) 13.25/4.41 3(4(0(1(1(5(3(5(0(4(4(3(4(3(x1)))))))))))))) -> 3(5(5(4(0(1(1(1(1(4(4(0(0(3(x1)))))))))))))) 13.25/4.41 4(0(0(2(4(5(4(2(1(4(3(4(4(3(x1)))))))))))))) -> 4(4(0(1(5(2(2(1(1(0(1(3(1(3(x1)))))))))))))) 13.25/4.41 1(5(2(4(4(0(2(4(5(5(1(2(3(3(1(x1))))))))))))))) -> 3(1(2(4(5(4(0(3(2(1(5(5(1(3(1(x1))))))))))))))) 13.25/4.41 2(0(2(0(5(5(5(1(4(4(4(3(3(3(0(x1))))))))))))))) -> 5(3(3(3(2(5(0(5(4(4(5(3(3(0(x1)))))))))))))) 13.25/4.41 5(0(4(2(1(1(2(4(1(0(3(1(0(3(5(x1))))))))))))))) -> 4(5(3(2(3(1(1(0(1(3(5(3(0(1(x1)))))))))))))) 13.25/4.41 5(0(4(5(4(0(4(0(4(5(2(0(1(2(1(x1))))))))))))))) -> 1(2(2(2(0(4(0(5(1(3(4(4(1(1(4(x1))))))))))))))) 13.25/4.41 5(3(1(2(3(2(2(2(4(1(0(4(3(3(4(0(x1)))))))))))))))) -> 5(5(4(0(2(5(0(0(1(3(0(3(4(3(4(0(x1)))))))))))))))) 13.25/4.41 1(3(4(1(3(3(0(2(3(3(3(2(5(4(4(4(4(x1))))))))))))))))) -> 1(1(3(2(5(0(4(2(4(3(4(0(3(3(3(0(4(x1))))))))))))))))) 13.25/4.41 3(4(3(3(4(3(2(4(1(1(2(1(4(5(3(0(0(x1))))))))))))))))) -> 1(1(3(0(5(3(2(0(4(4(1(3(5(0(0(2(0(x1))))))))))))))))) 13.25/4.41 5(3(3(4(5(3(5(1(2(5(5(0(5(5(1(5(3(x1))))))))))))))))) -> 5(5(3(5(2(1(3(3(5(1(3(5(0(5(4(3(x1)))))))))))))))) 13.25/4.41 2(0(1(2(0(3(5(1(4(5(2(3(5(2(4(1(0(3(x1)))))))))))))))))) -> 1(4(5(4(0(5(5(1(3(0(1(4(4(0(3(3(4(3(x1)))))))))))))))))) 13.25/4.41 5(1(1(0(2(2(1(1(5(5(2(2(4(0(4(3(2(2(x1)))))))))))))))))) -> 3(1(1(1(2(2(1(5(4(3(0(2(3(5(4(2(2(x1))))))))))))))))) 13.25/4.41 2(2(3(0(5(2(3(4(4(0(3(3(0(2(0(2(4(1(1(x1))))))))))))))))))) -> 2(1(5(2(1(1(5(4(5(4(3(3(5(4(5(5(3(4(x1)))))))))))))))))) 13.25/4.41 1(3(5(3(3(3(4(0(1(2(3(3(0(4(3(0(3(4(4(5(x1)))))))))))))))))))) -> 4(2(4(5(4(5(1(4(4(5(5(0(2(1(1(1(1(4(2(5(1(x1))))))))))))))))))))) 13.25/4.41 5(0(4(2(0(0(0(3(3(1(5(2(2(5(1(3(4(3(0(5(1(x1))))))))))))))))))))) -> 0(3(1(5(4(3(1(1(3(0(3(2(0(0(2(2(2(1(4(4(1(x1))))))))))))))))))))) 13.25/4.41 13.25/4.41 13.25/4.41 13.25/4.41 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (2) 13.25/4.41 Obligation: 13.25/4.41 Q restricted rewrite system: 13.25/4.41 The TRS R consists of the following rules: 13.25/4.41 13.25/4.41 3(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 4(3(5(2(3(3(1(3(0(3(x1)))))))))) 13.25/4.41 13.25/4.41 Q is empty. 13.25/4.41 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (3) DependencyPairsProof (EQUIVALENT) 13.25/4.41 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (4) 13.25/4.41 Obligation: 13.25/4.41 Q DP problem: 13.25/4.41 The TRS P consists of the following rules: 13.25/4.41 13.25/4.41 3^1(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 3^1(5(2(3(3(1(3(0(3(x1))))))))) 13.25/4.41 3^1(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 3^1(3(1(3(0(3(x1)))))) 13.25/4.41 3^1(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 3^1(1(3(0(3(x1))))) 13.25/4.41 3^1(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 3^1(0(3(x1))) 13.25/4.41 13.25/4.41 The TRS R consists of the following rules: 13.25/4.41 13.25/4.41 3(3(3(5(3(0(2(1(4(3(x1)))))))))) -> 4(3(5(2(3(3(1(3(0(3(x1)))))))))) 13.25/4.41 13.25/4.41 Q is empty. 13.25/4.41 We have to consider all minimal (P,Q,R)-chains. 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (5) DependencyGraphProof (EQUIVALENT) 13.25/4.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 13.25/4.41 ---------------------------------------- 13.25/4.41 13.25/4.41 (6) 13.25/4.41 TRUE 13.68/4.86 EOF