13.36/4.26 YES 16.09/4.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 16.09/4.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 16.09/4.59 16.09/4.59 16.09/4.59 Termination w.r.t. Q of the given QTRS could be proven: 16.09/4.59 16.09/4.59 (0) QTRS 16.09/4.59 (1) QTRSRRRProof [EQUIVALENT, 138 ms] 16.09/4.59 (2) QTRS 16.09/4.59 (3) Overlay + Local Confluence [EQUIVALENT, 1 ms] 16.09/4.59 (4) QTRS 16.09/4.59 (5) DependencyPairsProof [EQUIVALENT, 22 ms] 16.09/4.59 (6) QDP 16.09/4.59 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 16.09/4.59 (8) TRUE 16.09/4.59 16.09/4.59 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (0) 16.09/4.59 Obligation: 16.09/4.59 Q restricted rewrite system: 16.09/4.59 The TRS R consists of the following rules: 16.09/4.59 16.09/4.59 0(x1) -> 1(x1) 16.09/4.59 0(0(x1)) -> 0(x1) 16.09/4.59 3(4(5(x1))) -> 4(3(5(x1))) 16.09/4.59 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(0(1(0(1(1(0(1(0(0(1(0(1(0(1(1(0(1(1(0(1(0(1(0(1(0(1(1(0(0(0(1(1(1(0(1(0(0(0(0(1(1(0(1(0(1(0(1(0(1(1(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(0(1(1(0(0(0(0(1(1(1(1(0(1(0(0(0(1(0(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(0(1(0(0(1(0(0(0(0(0(1(1(1(1(0(0(1(1(1(0(1(1(0(1(1(1(1(1(0(0(1(1(0(0(1(0(0(0(0(1(0(1(0(0(0(0(1(1(0(1(0(0(1(1(1(0(1(0(0(0(1(1(0(1(0(0(1(0(0(1(1(1(1(1(1(0(1(0(0(1(0(1(0(1(1(0(1(1(0(0(1(1(1(1(0(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(0(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 16.09/4.59 0(1(1(0(1(0(0(0(1(1(1(1(1(1(1(1(1(1(1(0(0(1(1(0(1(1(1(0(1(0(1(1(0(1(0(1(1(0(0(0(1(1(0(1(1(1(0(1(0(1(0(0(0(0(0(1(0(0(1(0(1(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(0(1(1(0(1(1(1(1(1(1(1(1(1(0(0(1(0(0(0(0(1(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(1(0(0(1(1(0(0(0(1(0(0(0(0(1(1(1(1(0(1(1(1(1(0(0(0(1(1(0(1(0(1(0(0(0(0(1(0(0(1(1(1(0(1(0(0(0(0(0(0(0(1(0(1(0(0(1(0(0(0(0(0(0(0(0(1(0(1(1(0(1(1(1(1(0(1(0(0(0(0(0(1(1(1(1(0(0(0(0(0(0(1(0(0(1(1(1(1(1(0(0(0(1(1(1(0(0(1(1(1(1(1(0(0(1(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 16.09/4.59 16.09/4.59 Q is empty. 16.09/4.59 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (1) QTRSRRRProof (EQUIVALENT) 16.09/4.59 Used ordering: 16.09/4.59 Polynomial interpretation [POLO]: 16.09/4.59 16.09/4.59 POL(0(x_1)) = 11 + x_1 16.09/4.59 POL(1(x_1)) = x_1 16.09/4.59 POL(2(x_1)) = 7 + x_1 16.09/4.59 POL(3(x_1)) = x_1 16.09/4.59 POL(4(x_1)) = x_1 16.09/4.59 POL(5(x_1)) = x_1 16.09/4.59 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 16.09/4.59 16.09/4.59 0(x1) -> 1(x1) 16.09/4.59 0(0(x1)) -> 0(x1) 16.09/4.59 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 0(0(1(0(1(1(0(1(0(0(1(0(1(0(1(1(0(1(1(0(1(0(1(0(1(0(1(1(0(0(0(1(1(1(0(1(0(0(0(0(1(1(0(1(0(1(0(1(0(1(1(0(1(0(1(1(1(0(1(0(1(1(1(0(0(0(1(0(0(1(1(0(0(0(0(1(1(1(1(0(1(0(0(0(1(0(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(1(0(0(1(0(0(1(0(1(1(0(1(0(0(1(0(0(0(0(0(1(1(1(1(0(0(1(1(1(0(1(1(0(1(1(1(1(1(0(0(1(1(0(0(1(0(0(0(0(1(0(1(0(0(0(0(1(1(0(1(0(0(1(1(1(0(1(0(0(0(1(1(0(1(0(0(1(0(0(1(1(1(1(1(1(0(1(0(0(1(0(1(0(1(1(0(1(1(0(0(1(1(1(1(0(0(1(1(1(0(1(1(0(1(0(0(0(1(0(1(0(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 16.09/4.59 0(1(1(0(1(0(0(0(1(1(1(1(1(1(1(1(1(1(1(0(0(1(1(0(1(1(1(0(1(0(1(1(0(1(0(1(1(0(0(0(1(1(0(1(1(1(0(1(0(1(0(0(0(0(0(1(0(0(1(0(1(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(0(1(1(0(1(1(1(1(1(1(1(1(1(0(0(1(0(0(0(0(1(0(0(1(0(0(0(1(0(0(1(0(1(1(1(1(1(0(0(1(1(0(0(0(1(0(0(0(0(1(1(1(1(0(1(1(1(1(0(0(0(1(1(0(1(0(1(0(0(0(0(1(0(0(1(1(1(0(1(0(0(0(0(0(0(0(1(0(1(0(0(1(0(0(0(0(0(0(0(0(1(0(1(1(0(1(1(1(1(0(1(0(0(0(0(0(1(1(1(1(0(0(0(0(0(0(1(0(0(1(1(1(1(1(0(0(0(1(1(1(0(0(1(1(1(1(1(0(0(1(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 16.09/4.59 16.09/4.59 16.09/4.59 16.09/4.59 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (2) 16.09/4.59 Obligation: 16.09/4.59 Q restricted rewrite system: 16.09/4.59 The TRS R consists of the following rules: 16.09/4.59 16.09/4.59 3(4(5(x1))) -> 4(3(5(x1))) 16.09/4.59 16.09/4.59 Q is empty. 16.09/4.59 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (3) Overlay + Local Confluence (EQUIVALENT) 16.09/4.59 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (4) 16.09/4.59 Obligation: 16.09/4.59 Q restricted rewrite system: 16.09/4.59 The TRS R consists of the following rules: 16.09/4.59 16.09/4.59 3(4(5(x1))) -> 4(3(5(x1))) 16.09/4.59 16.09/4.59 The set Q consists of the following terms: 16.09/4.59 16.09/4.59 3(4(5(x0))) 16.09/4.59 16.09/4.59 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (5) DependencyPairsProof (EQUIVALENT) 16.09/4.59 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (6) 16.09/4.59 Obligation: 16.09/4.59 Q DP problem: 16.09/4.59 The TRS P consists of the following rules: 16.09/4.59 16.09/4.59 3^1(4(5(x1))) -> 3^1(5(x1)) 16.09/4.59 16.09/4.59 The TRS R consists of the following rules: 16.09/4.59 16.09/4.59 3(4(5(x1))) -> 4(3(5(x1))) 16.09/4.59 16.09/4.59 The set Q consists of the following terms: 16.09/4.59 16.09/4.59 3(4(5(x0))) 16.09/4.59 16.09/4.59 We have to consider all minimal (P,Q,R)-chains. 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (7) DependencyGraphProof (EQUIVALENT) 16.09/4.59 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 16.09/4.59 ---------------------------------------- 16.09/4.59 16.09/4.59 (8) 16.09/4.59 TRUE 16.33/4.67 EOF