10.13/3.36 YES 10.88/3.54 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 10.88/3.54 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.88/3.54 10.88/3.54 10.88/3.54 Termination w.r.t. Q of the given QTRS could be proven: 10.88/3.54 10.88/3.54 (0) QTRS 10.88/3.54 (1) QTRSRRRProof [EQUIVALENT, 187 ms] 10.88/3.54 (2) QTRS 10.88/3.54 (3) AAECC Innermost [EQUIVALENT, 0 ms] 10.88/3.54 (4) QTRS 10.88/3.54 (5) DependencyPairsProof [EQUIVALENT, 14 ms] 10.88/3.54 (6) QDP 10.88/3.54 (7) DependencyGraphProof [EQUIVALENT, 1 ms] 10.88/3.54 (8) TRUE 10.88/3.54 10.88/3.54 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (0) 10.88/3.54 Obligation: 10.88/3.54 Q restricted rewrite system: 10.88/3.54 The TRS R consists of the following rules: 10.88/3.54 10.88/3.54 0(x1) -> 1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 10.88/3.54 2(x1) -> 3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 10.88/3.54 0(1(2(3(4(5(4(5(x1)))))))) -> 0(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 Q is empty. 10.88/3.54 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (1) QTRSRRRProof (EQUIVALENT) 10.88/3.54 Used ordering: 10.88/3.54 Polynomial interpretation [POLO]: 10.88/3.54 10.88/3.54 POL(0(x_1)) = 1 + x_1 10.88/3.54 POL(1(x_1)) = x_1 10.88/3.54 POL(2(x_1)) = 1 + x_1 10.88/3.54 POL(3(x_1)) = x_1 10.88/3.54 POL(4(x_1)) = x_1 10.88/3.54 POL(5(x_1)) = x_1 10.88/3.54 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 10.88/3.54 10.88/3.54 0(x1) -> 1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 10.88/3.54 2(x1) -> 3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(3(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 10.88/3.54 10.88/3.54 10.88/3.54 10.88/3.54 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (2) 10.88/3.54 Obligation: 10.88/3.54 Q restricted rewrite system: 10.88/3.54 The TRS R consists of the following rules: 10.88/3.54 10.88/3.54 0(1(2(3(4(5(4(5(x1)))))))) -> 0(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 Q is empty. 10.88/3.54 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (3) AAECC Innermost (EQUIVALENT) 10.88/3.54 We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none 10.88/3.54 10.88/3.54 The TRS R 2 is 10.88/3.54 0(1(2(3(4(5(4(5(x1)))))))) -> 0(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 The signature Sigma is {0_1} 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (4) 10.88/3.54 Obligation: 10.88/3.54 Q restricted rewrite system: 10.88/3.54 The TRS R consists of the following rules: 10.88/3.54 10.88/3.54 0(1(2(3(4(5(4(5(x1)))))))) -> 0(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 The set Q consists of the following terms: 10.88/3.54 10.88/3.54 0(1(2(3(4(5(4(5(x0)))))))) 10.88/3.54 10.88/3.54 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (5) DependencyPairsProof (EQUIVALENT) 10.88/3.54 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (6) 10.88/3.54 Obligation: 10.88/3.54 Q DP problem: 10.88/3.54 The TRS P consists of the following rules: 10.88/3.54 10.88/3.54 0^1(1(2(3(4(5(4(5(x1)))))))) -> 0^1(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 The TRS R consists of the following rules: 10.88/3.54 10.88/3.54 0(1(2(3(4(5(4(5(x1)))))))) -> 0(1(2(3(4(4(5(5(x1)))))))) 10.88/3.54 10.88/3.54 The set Q consists of the following terms: 10.88/3.54 10.88/3.54 0(1(2(3(4(5(4(5(x0)))))))) 10.88/3.54 10.88/3.54 We have to consider all minimal (P,Q,R)-chains. 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (7) DependencyGraphProof (EQUIVALENT) 10.88/3.54 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 10.88/3.54 ---------------------------------------- 10.88/3.54 10.88/3.54 (8) 10.88/3.54 TRUE 11.14/3.71 EOF