9.90/3.42 YES 10.14/3.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 10.14/3.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.14/3.44 10.14/3.44 10.14/3.44 Termination w.r.t. Q of the given QTRS could be proven: 10.14/3.44 10.14/3.44 (0) QTRS 10.14/3.44 (1) QTRSRRRProof [EQUIVALENT, 136 ms] 10.14/3.44 (2) QTRS 10.14/3.44 (3) Overlay + Local Confluence [EQUIVALENT, 0 ms] 10.14/3.44 (4) QTRS 10.14/3.44 (5) DependencyPairsProof [EQUIVALENT, 12 ms] 10.14/3.44 (6) QDP 10.14/3.44 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 10.14/3.44 (8) TRUE 10.14/3.44 10.14/3.44 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (0) 10.14/3.44 Obligation: 10.14/3.44 Q restricted rewrite system: 10.14/3.44 The TRS R consists of the following rules: 10.14/3.44 10.14/3.44 0(1(2(2(3(0(4(4(x1)))))))) -> 0(4(5(0(3(4(0(4(x1)))))))) 10.14/3.44 0(2(2(2(0(5(2(5(4(x1))))))))) -> 0(0(0(3(4(3(0(1(0(x1))))))))) 10.14/3.44 2(1(1(3(4(3(1(1(5(x1))))))))) -> 1(0(2(3(1(0(5(1(5(x1))))))))) 10.14/3.44 5(4(0(4(3(3(1(2(5(3(0(x1))))))))))) -> 5(5(2(5(3(1(5(0(3(5(2(x1))))))))))) 10.14/3.44 2(4(5(0(1(1(3(3(5(3(0(0(x1)))))))))))) -> 4(4(2(2(1(0(4(0(1(3(2(0(x1)))))))))))) 10.14/3.44 4(4(3(0(3(1(5(3(5(1(3(1(5(3(x1)))))))))))))) -> 0(2(4(5(0(0(0(5(1(0(5(4(4(x1))))))))))))) 10.14/3.44 2(2(0(0(2(1(0(5(3(2(2(1(4(0(5(x1))))))))))))))) -> 0(0(3(5(3(0(4(3(1(3(0(2(5(5(x1)))))))))))))) 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3(4(0(1(0(5(5(3(1(4(0(3(5(3(2(2(5(x1))))))))))))))))) 10.14/3.44 5(1(1(4(1(5(3(0(4(3(2(5(4(1(3(3(5(x1))))))))))))))))) -> 5(0(1(0(4(0(2(4(5(1(5(4(1(5(3(3(5(x1))))))))))))))))) 10.14/3.44 0(1(4(4(3(2(0(4(1(4(3(4(4(1(5(3(4(4(x1)))))))))))))))))) -> 0(2(4(2(3(1(0(1(1(1(3(0(2(4(4(1(1(2(x1)))))))))))))))))) 10.14/3.44 1(1(4(1(0(1(0(3(3(4(4(1(5(4(0(4(4(5(5(3(x1)))))))))))))))))))) -> 1(3(0(3(2(2(4(4(2(0(3(3(4(0(3(0(4(3(4(0(x1)))))))))))))))))))) 10.14/3.44 2(0(3(3(3(4(1(1(0(4(4(0(3(3(3(0(0(1(5(3(x1)))))))))))))))))))) -> 5(3(3(1(4(0(4(5(4(4(4(2(4(3(1(1(1(5(4(x1))))))))))))))))))) 10.14/3.44 2(3(3(2(1(5(0(5(0(1(3(3(2(5(1(5(0(3(0(5(x1)))))))))))))))))))) -> 3(4(0(2(5(5(2(4(2(4(3(1(1(4(4(5(5(3(5(x1))))))))))))))))))) 10.14/3.44 3(3(2(2(3(3(4(0(0(0(2(5(0(5(3(0(0(1(1(4(x1)))))))))))))))))))) -> 3(1(1(0(3(3(5(4(2(2(1(0(1(1(0(5(0(3(4(0(x1)))))))))))))))))))) 10.14/3.44 4(2(4(1(0(5(0(4(1(0(3(0(2(5(4(3(5(3(5(3(x1)))))))))))))))))))) -> 4(2(5(4(4(2(4(5(0(0(3(1(5(0(2(0(2(2(1(x1))))))))))))))))))) 10.14/3.44 10.14/3.44 Q is empty. 10.14/3.44 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (1) QTRSRRRProof (EQUIVALENT) 10.14/3.44 Used ordering: 10.14/3.44 Polynomial interpretation [POLO]: 10.14/3.44 10.14/3.44 POL(0(x_1)) = 104 + x_1 10.14/3.44 POL(1(x_1)) = 131 + x_1 10.14/3.44 POL(2(x_1)) = 126 + x_1 10.14/3.44 POL(3(x_1)) = 164 + x_1 10.14/3.44 POL(4(x_1)) = 138 + x_1 10.14/3.44 POL(5(x_1)) = 139 + x_1 10.14/3.44 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 10.14/3.44 10.14/3.44 0(1(2(2(3(0(4(4(x1)))))))) -> 0(4(5(0(3(4(0(4(x1)))))))) 10.14/3.44 0(2(2(2(0(5(2(5(4(x1))))))))) -> 0(0(0(3(4(3(0(1(0(x1))))))))) 10.14/3.44 2(1(1(3(4(3(1(1(5(x1))))))))) -> 1(0(2(3(1(0(5(1(5(x1))))))))) 10.14/3.44 5(4(0(4(3(3(1(2(5(3(0(x1))))))))))) -> 5(5(2(5(3(1(5(0(3(5(2(x1))))))))))) 10.14/3.44 2(4(5(0(1(1(3(3(5(3(0(0(x1)))))))))))) -> 4(4(2(2(1(0(4(0(1(3(2(0(x1)))))))))))) 10.14/3.44 4(4(3(0(3(1(5(3(5(1(3(1(5(3(x1)))))))))))))) -> 0(2(4(5(0(0(0(5(1(0(5(4(4(x1))))))))))))) 10.14/3.44 2(2(0(0(2(1(0(5(3(2(2(1(4(0(5(x1))))))))))))))) -> 0(0(3(5(3(0(4(3(1(3(0(2(5(5(x1)))))))))))))) 10.14/3.44 5(1(1(4(1(5(3(0(4(3(2(5(4(1(3(3(5(x1))))))))))))))))) -> 5(0(1(0(4(0(2(4(5(1(5(4(1(5(3(3(5(x1))))))))))))))))) 10.14/3.44 0(1(4(4(3(2(0(4(1(4(3(4(4(1(5(3(4(4(x1)))))))))))))))))) -> 0(2(4(2(3(1(0(1(1(1(3(0(2(4(4(1(1(2(x1)))))))))))))))))) 10.14/3.44 1(1(4(1(0(1(0(3(3(4(4(1(5(4(0(4(4(5(5(3(x1)))))))))))))))))))) -> 1(3(0(3(2(2(4(4(2(0(3(3(4(0(3(0(4(3(4(0(x1)))))))))))))))))))) 10.14/3.44 2(0(3(3(3(4(1(1(0(4(4(0(3(3(3(0(0(1(5(3(x1)))))))))))))))))))) -> 5(3(3(1(4(0(4(5(4(4(4(2(4(3(1(1(1(5(4(x1))))))))))))))))))) 10.14/3.44 2(3(3(2(1(5(0(5(0(1(3(3(2(5(1(5(0(3(0(5(x1)))))))))))))))))))) -> 3(4(0(2(5(5(2(4(2(4(3(1(1(4(4(5(5(3(5(x1))))))))))))))))))) 10.14/3.44 3(3(2(2(3(3(4(0(0(0(2(5(0(5(3(0(0(1(1(4(x1)))))))))))))))))))) -> 3(1(1(0(3(3(5(4(2(2(1(0(1(1(0(5(0(3(4(0(x1)))))))))))))))))))) 10.14/3.44 4(2(4(1(0(5(0(4(1(0(3(0(2(5(4(3(5(3(5(3(x1)))))))))))))))))))) -> 4(2(5(4(4(2(4(5(0(0(3(1(5(0(2(0(2(2(1(x1))))))))))))))))))) 10.14/3.44 10.14/3.44 10.14/3.44 10.14/3.44 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (2) 10.14/3.44 Obligation: 10.14/3.44 Q restricted rewrite system: 10.14/3.44 The TRS R consists of the following rules: 10.14/3.44 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3(4(0(1(0(5(5(3(1(4(0(3(5(3(2(2(5(x1))))))))))))))))) 10.14/3.44 10.14/3.44 Q is empty. 10.14/3.44 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (3) Overlay + Local Confluence (EQUIVALENT) 10.14/3.44 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (4) 10.14/3.44 Obligation: 10.14/3.44 Q restricted rewrite system: 10.14/3.44 The TRS R consists of the following rules: 10.14/3.44 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3(4(0(1(0(5(5(3(1(4(0(3(5(3(2(2(5(x1))))))))))))))))) 10.14/3.44 10.14/3.44 The set Q consists of the following terms: 10.14/3.44 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x0))))))))))))))))) 10.14/3.44 10.14/3.44 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (5) DependencyPairsProof (EQUIVALENT) 10.14/3.44 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (6) 10.14/3.44 Obligation: 10.14/3.44 Q DP problem: 10.14/3.44 The TRS P consists of the following rules: 10.14/3.44 10.14/3.44 3^1(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3^1(4(0(1(0(5(5(3(1(4(0(3(5(3(2(2(5(x1))))))))))))))))) 10.14/3.44 3^1(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3^1(1(4(0(3(5(3(2(2(5(x1)))))))))) 10.14/3.44 3^1(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3^1(5(3(2(2(5(x1)))))) 10.14/3.44 3^1(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3^1(2(2(5(x1)))) 10.14/3.44 10.14/3.44 The TRS R consists of the following rules: 10.14/3.44 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x1))))))))))))))))) -> 3(4(0(1(0(5(5(3(1(4(0(3(5(3(2(2(5(x1))))))))))))))))) 10.14/3.44 10.14/3.44 The set Q consists of the following terms: 10.14/3.44 10.14/3.44 3(0(0(4(2(5(5(1(3(0(2(3(3(5(1(4(5(x0))))))))))))))))) 10.14/3.44 10.14/3.44 We have to consider all minimal (P,Q,R)-chains. 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (7) DependencyGraphProof (EQUIVALENT) 10.14/3.44 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 10.14/3.44 ---------------------------------------- 10.14/3.44 10.14/3.44 (8) 10.14/3.44 TRUE 10.27/3.53 EOF