10.02/3.44 YES 10.74/3.66 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 10.74/3.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.74/3.66 10.74/3.66 10.74/3.66 Termination w.r.t. Q of the given QTRS could be proven: 10.74/3.66 10.74/3.66 (0) QTRS 10.74/3.66 (1) QTRSRRRProof [EQUIVALENT, 156 ms] 10.74/3.66 (2) QTRS 10.74/3.66 (3) DependencyPairsProof [EQUIVALENT, 28 ms] 10.74/3.66 (4) QDP 10.74/3.66 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 10.74/3.66 (6) TRUE 10.74/3.66 10.74/3.66 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (0) 10.74/3.66 Obligation: 10.74/3.66 Q restricted rewrite system: 10.74/3.66 The TRS R consists of the following rules: 10.74/3.66 10.74/3.66 0(1(0(2(x1)))) -> 2(2(2(x1))) 10.74/3.66 0(1(1(1(x1)))) -> 3(1(0(x1))) 10.74/3.66 4(5(1(4(x1)))) -> 1(5(4(4(x1)))) 10.74/3.66 1(2(0(4(1(x1))))) -> 2(5(2(2(x1)))) 10.74/3.66 1(5(1(0(3(x1))))) -> 5(5(0(3(x1)))) 10.74/3.66 1(3(5(1(0(4(x1)))))) -> 4(0(0(0(1(4(x1)))))) 10.74/3.66 3(3(1(4(5(4(5(5(x1)))))))) -> 3(5(3(2(0(5(2(x1))))))) 10.74/3.66 4(2(5(3(1(0(0(5(2(x1))))))))) -> 4(4(1(5(3(5(3(2(x1)))))))) 10.74/3.66 2(2(1(1(0(2(0(1(2(3(x1)))))))))) -> 5(4(4(5(2(5(5(2(3(x1))))))))) 10.74/3.66 2(2(2(3(3(4(3(3(5(4(5(x1))))))))))) -> 5(5(4(2(5(3(5(3(2(0(5(x1))))))))))) 10.74/3.66 4(2(1(1(5(1(0(0(0(2(5(x1))))))))))) -> 5(5(0(3(0(1(4(5(2(5(x1)))))))))) 10.74/3.66 5(4(5(4(2(0(2(2(3(0(0(2(x1)))))))))))) -> 5(3(4(5(0(3(2(1(1(2(2(x1))))))))))) 10.74/3.66 0(2(1(1(3(1(5(0(5(2(3(1(4(x1))))))))))))) -> 3(0(1(5(1(4(4(5(0(5(1(1(4(x1))))))))))))) 10.74/3.66 1(1(2(4(2(2(2(1(1(3(4(1(2(x1))))))))))))) -> 1(5(0(3(4(1(0(5(2(2(2(3(x1)))))))))))) 10.74/3.66 1(4(0(0(2(4(0(3(3(2(0(3(1(x1))))))))))))) -> 1(5(0(5(0(2(0(5(1(0(1(2(5(5(x1)))))))))))))) 10.74/3.66 4(1(4(1(3(0(0(2(1(5(4(1(0(0(x1)))))))))))))) -> 2(3(0(2(5(4(4(3(0(4(1(2(0(x1))))))))))))) 10.74/3.66 4(2(4(4(0(0(0(4(0(3(5(0(3(3(x1)))))))))))))) -> 5(1(0(1(0(1(3(5(3(5(5(0(0(3(x1)))))))))))))) 10.74/3.66 1(2(5(3(3(0(2(2(5(3(2(3(3(3(2(1(x1)))))))))))))))) -> 1(1(0(0(4(1(2(1(0(5(0(0(3(5(0(4(1(x1))))))))))))))))) 10.74/3.66 3(1(2(5(0(3(4(3(1(5(4(1(5(2(0(5(x1)))))))))))))))) -> 3(1(0(4(2(0(5(2(4(4(2(2(1(1(1(5(x1)))))))))))))))) 10.74/3.66 1(3(4(5(1(0(0(3(1(2(4(2(3(5(2(0(4(1(x1)))))))))))))))))) -> 1(3(0(0(4(2(2(5(3(1(0(1(2(1(5(0(1(4(1(x1))))))))))))))))))) 10.74/3.66 1(0(3(3(2(5(0(0(3(0(3(2(4(1(4(0(2(4(2(x1))))))))))))))))))) -> 4(3(4(2(1(4(2(4(3(3(2(2(2(1(1(1(5(3(2(x1))))))))))))))))))) 10.74/3.66 5(3(2(1(0(1(3(1(3(3(0(0(3(2(5(3(0(3(0(x1))))))))))))))))))) -> 5(1(0(3(2(4(0(0(3(2(1(5(1(3(0(5(1(3(1(x1))))))))))))))))))) 10.74/3.66 0(2(2(3(3(5(0(0(5(0(3(1(3(0(1(2(1(5(5(1(x1)))))))))))))))))))) -> 0(0(3(2(5(2(5(2(3(2(1(2(5(4(3(4(5(0(4(x1))))))))))))))))))) 10.74/3.66 1(0(4(2(3(3(5(4(3(5(0(2(0(4(5(0(2(0(2(4(x1)))))))))))))))))))) -> 4(5(4(5(4(1(1(2(5(0(4(3(1(5(4(3(1(5(4(0(4(x1))))))))))))))))))))) 10.74/3.66 1(4(2(3(1(3(4(2(4(1(5(1(4(0(4(5(2(0(0(3(4(x1))))))))))))))))))))) -> 4(3(4(1(3(5(1(1(4(3(1(5(1(3(1(2(4(2(1(3(x1)))))))))))))))))))) 10.74/3.66 10.74/3.66 Q is empty. 10.74/3.66 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (1) QTRSRRRProof (EQUIVALENT) 10.74/3.66 Used ordering: 10.74/3.66 Polynomial interpretation [POLO]: 10.74/3.66 10.74/3.66 POL(0(x_1)) = 235 + x_1 10.74/3.66 POL(1(x_1)) = 191 + x_1 10.74/3.66 POL(2(x_1)) = 157 + x_1 10.74/3.66 POL(3(x_1)) = 381 + x_1 10.74/3.66 POL(4(x_1)) = 237 + x_1 10.74/3.66 POL(5(x_1)) = 136 + x_1 10.74/3.66 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 10.74/3.66 10.74/3.66 0(1(0(2(x1)))) -> 2(2(2(x1))) 10.74/3.66 0(1(1(1(x1)))) -> 3(1(0(x1))) 10.74/3.66 1(2(0(4(1(x1))))) -> 2(5(2(2(x1)))) 10.74/3.66 1(5(1(0(3(x1))))) -> 5(5(0(3(x1)))) 10.74/3.66 1(3(5(1(0(4(x1)))))) -> 4(0(0(0(1(4(x1)))))) 10.74/3.66 3(3(1(4(5(4(5(5(x1)))))))) -> 3(5(3(2(0(5(2(x1))))))) 10.74/3.66 4(2(5(3(1(0(0(5(2(x1))))))))) -> 4(4(1(5(3(5(3(2(x1)))))))) 10.74/3.66 2(2(1(1(0(2(0(1(2(3(x1)))))))))) -> 5(4(4(5(2(5(5(2(3(x1))))))))) 10.74/3.66 2(2(2(3(3(4(3(3(5(4(5(x1))))))))))) -> 5(5(4(2(5(3(5(3(2(0(5(x1))))))))))) 10.74/3.66 4(2(1(1(5(1(0(0(0(2(5(x1))))))))))) -> 5(5(0(3(0(1(4(5(2(5(x1)))))))))) 10.74/3.66 5(4(5(4(2(0(2(2(3(0(0(2(x1)))))))))))) -> 5(3(4(5(0(3(2(1(1(2(2(x1))))))))))) 10.74/3.66 0(2(1(1(3(1(5(0(5(2(3(1(4(x1))))))))))))) -> 3(0(1(5(1(4(4(5(0(5(1(1(4(x1))))))))))))) 10.74/3.66 1(1(2(4(2(2(2(1(1(3(4(1(2(x1))))))))))))) -> 1(5(0(3(4(1(0(5(2(2(2(3(x1)))))))))))) 10.74/3.66 1(4(0(0(2(4(0(3(3(2(0(3(1(x1))))))))))))) -> 1(5(0(5(0(2(0(5(1(0(1(2(5(5(x1)))))))))))))) 10.74/3.66 4(1(4(1(3(0(0(2(1(5(4(1(0(0(x1)))))))))))))) -> 2(3(0(2(5(4(4(3(0(4(1(2(0(x1))))))))))))) 10.74/3.66 4(2(4(4(0(0(0(4(0(3(5(0(3(3(x1)))))))))))))) -> 5(1(0(1(0(1(3(5(3(5(5(0(0(3(x1)))))))))))))) 10.74/3.66 1(2(5(3(3(0(2(2(5(3(2(3(3(3(2(1(x1)))))))))))))))) -> 1(1(0(0(4(1(2(1(0(5(0(0(3(5(0(4(1(x1))))))))))))))))) 10.74/3.66 3(1(2(5(0(3(4(3(1(5(4(1(5(2(0(5(x1)))))))))))))))) -> 3(1(0(4(2(0(5(2(4(4(2(2(1(1(1(5(x1)))))))))))))))) 10.74/3.66 1(3(4(5(1(0(0(3(1(2(4(2(3(5(2(0(4(1(x1)))))))))))))))))) -> 1(3(0(0(4(2(2(5(3(1(0(1(2(1(5(0(1(4(1(x1))))))))))))))))))) 10.74/3.66 1(0(3(3(2(5(0(0(3(0(3(2(4(1(4(0(2(4(2(x1))))))))))))))))))) -> 4(3(4(2(1(4(2(4(3(3(2(2(2(1(1(1(5(3(2(x1))))))))))))))))))) 10.74/3.66 5(3(2(1(0(1(3(1(3(3(0(0(3(2(5(3(0(3(0(x1))))))))))))))))))) -> 5(1(0(3(2(4(0(0(3(2(1(5(1(3(0(5(1(3(1(x1))))))))))))))))))) 10.74/3.66 0(2(2(3(3(5(0(0(5(0(3(1(3(0(1(2(1(5(5(1(x1)))))))))))))))))))) -> 0(0(3(2(5(2(5(2(3(2(1(2(5(4(3(4(5(0(4(x1))))))))))))))))))) 10.74/3.66 1(0(4(2(3(3(5(4(3(5(0(2(0(4(5(0(2(0(2(4(x1)))))))))))))))))))) -> 4(5(4(5(4(1(1(2(5(0(4(3(1(5(4(3(1(5(4(0(4(x1))))))))))))))))))))) 10.74/3.66 1(4(2(3(1(3(4(2(4(1(5(1(4(0(4(5(2(0(0(3(4(x1))))))))))))))))))))) -> 4(3(4(1(3(5(1(1(4(3(1(5(1(3(1(2(4(2(1(3(x1)))))))))))))))))))) 10.74/3.66 10.74/3.66 10.74/3.66 10.74/3.66 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (2) 10.74/3.66 Obligation: 10.74/3.66 Q restricted rewrite system: 10.74/3.66 The TRS R consists of the following rules: 10.74/3.66 10.74/3.66 4(5(1(4(x1)))) -> 1(5(4(4(x1)))) 10.74/3.66 10.74/3.66 Q is empty. 10.74/3.66 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (3) DependencyPairsProof (EQUIVALENT) 10.74/3.66 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (4) 10.74/3.66 Obligation: 10.74/3.66 Q DP problem: 10.74/3.66 The TRS P consists of the following rules: 10.74/3.66 10.74/3.66 4^1(5(1(4(x1)))) -> 4^1(4(x1)) 10.74/3.66 10.74/3.66 The TRS R consists of the following rules: 10.74/3.66 10.74/3.66 4(5(1(4(x1)))) -> 1(5(4(4(x1)))) 10.74/3.66 10.74/3.66 Q is empty. 10.74/3.66 We have to consider all minimal (P,Q,R)-chains. 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (5) DependencyGraphProof (EQUIVALENT) 10.74/3.66 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 10.74/3.66 ---------------------------------------- 10.74/3.66 10.74/3.66 (6) 10.74/3.66 TRUE 11.04/3.78 EOF