10.66/3.51 YES 11.06/3.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 11.06/3.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.06/3.53 11.06/3.53 11.06/3.53 Termination w.r.t. Q of the given QTRS could be proven: 11.06/3.53 11.06/3.53 (0) QTRS 11.06/3.53 (1) QTRSRRRProof [EQUIVALENT, 167 ms] 11.06/3.53 (2) QTRS 11.06/3.53 (3) Overlay + Local Confluence [EQUIVALENT, 0 ms] 11.06/3.53 (4) QTRS 11.06/3.53 (5) DependencyPairsProof [EQUIVALENT, 0 ms] 11.06/3.53 (6) QDP 11.06/3.53 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 11.06/3.53 (8) TRUE 11.06/3.53 11.06/3.53 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (0) 11.06/3.53 Obligation: 11.06/3.53 Q restricted rewrite system: 11.06/3.53 The TRS R consists of the following rules: 11.06/3.53 11.06/3.53 0(x1) -> 1(x1) 11.06/3.53 0(0(x1)) -> 0(x1) 11.06/3.53 3(4(5(x1))) -> 4(3(5(x1))) 11.06/3.53 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(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(1(0(1(0(1(1(1(1(1(1(1(1(1(0(1(0(0(0(1(1(0(0(1(0(0(0(0(0(0(1(0(0(1(0(0(0(0(0(1(1(1(0(0(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(0(0(0(0(1(0(1(1(0(1(1(1(0(0(1(0(1(1(0(1(0(0(0(0(0(0(0(0(0(0(1(0(1(0(1(0(0(1(1(1(0(1(1(1(0(0(1(1(0(1(1(1(1(1(0(1(1(1(1(0(0(0(0(0(1(1(1(0(0(1(0(1(1(0(1(0(1(1(1(1(0(1(0(0(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(0(0(0(1(0(0(1(1(1(1(1(0(1(1(1(1(0(1(0(1(0(1(0(0(0(1(0(0(0(1(1(1(0(0(0(1(0(0(0(0(1(0(0(1(1(1(1(1(1(0(1(1(1(1(0(1(0(0(1(0(0(0(1(1(0(0(1(1(1(1(0(1(1(1(0(0(1(0(0(1(0(1(0(1(1(1(1(0(0(0(1(1(1(1(0(0(0(1(0(1(0(1(0(1(0(1(1(1(1(0(1(1(1(1(1(0(0(1(1(0(0(1(1(1(1(0(1(0(0(0(1(0(1(1(0(0(1(1(0(0(0(0(0(1(1(0(1(0(1(0(1(1(0(0(0(0(1(0(1(0(0(0(0(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 11.06/3.53 0(0(0(1(0(1(0(1(0(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(1(0(1(1(0(0(0(1(0(0(1(0(0(1(0(0(1(1(1(1(0(1(1(1(1(1(1(1(1(0(0(1(0(0(1(0(0(0(0(1(0(0(0(1(0(0(1(0(1(0(1(0(1(0(0(1(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(1(1(1(1(0(0(1(1(1(0(0(0(0(1(0(0(1(0(0(0(0(0(1(1(1(0(1(1(1(1(1(1(1(1(1(1(0(0(0(0(1(0(1(0(1(0(1(1(0(0(1(1(0(1(1(1(0(1(1(0(0(1(1(0(1(1(0(0(0(0(1(1(1(0(0(1(1(0(1(0(1(1(1(0(0(1(0(0(1(0(0(1(0(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(0(1(0(0(1(1(1(1(1(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(0(1(1(0(0(1(0(0(1(0(0(0(0(1(1(0(1(1(0(0(0(1(1(1(1(0(1(0(0(0(0(0(0(0(1(0(1(1(0(1(1(0(1(1(0(1(1(1(0(0(0(1(1(1(1(0(0(1(1(1(0(0(1(0(0(1(1(0(0(0(0(1(1(0(0(0(0(0(1(1(0(1(0(0(1(0(1(0(1(0(0(1(1(1(1(1(0(1(1(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(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 11.06/3.53 11.06/3.53 Q is empty. 11.06/3.53 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (1) QTRSRRRProof (EQUIVALENT) 11.06/3.53 Used ordering: 11.06/3.53 Polynomial interpretation [POLO]: 11.06/3.53 11.06/3.53 POL(0(x_1)) = 11 + x_1 11.06/3.53 POL(1(x_1)) = 10 + x_1 11.06/3.53 POL(2(x_1)) = 27 + x_1 11.06/3.53 POL(3(x_1)) = x_1 11.06/3.53 POL(4(x_1)) = x_1 11.06/3.53 POL(5(x_1)) = x_1 11.06/3.53 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 11.06/3.53 11.06/3.53 0(x1) -> 1(x1) 11.06/3.53 0(0(x1)) -> 0(x1) 11.06/3.53 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(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(1(0(1(0(1(1(1(1(1(1(1(1(1(0(1(0(0(0(1(1(0(0(1(0(0(0(0(0(0(1(0(0(1(0(0(0(0(0(1(1(1(0(0(0(1(1(0(0(1(1(1(0(0(1(0(1(0(0(0(1(0(0(0(0(1(0(1(1(0(1(1(1(0(0(1(0(1(1(0(1(0(0(0(0(0(0(0(0(0(0(1(0(1(0(1(0(0(1(1(1(0(1(1(1(0(0(1(1(0(1(1(1(1(1(0(1(1(1(1(0(0(0(0(0(1(1(1(0(0(1(0(1(1(0(1(0(1(1(1(1(0(1(0(0(0(0(1(0(1(1(0(1(1(0(0(1(1(0(1(0(0(0(1(0(0(1(1(1(1(1(0(1(1(1(1(0(1(0(1(0(1(0(0(0(1(0(0(0(1(1(1(0(0(0(1(0(0(0(0(1(0(0(1(1(1(1(1(1(0(1(1(1(1(0(1(0(0(1(0(0(0(1(1(0(0(1(1(1(1(0(1(1(1(0(0(1(0(0(1(0(1(0(1(1(1(1(0(0(0(1(1(1(1(0(0(0(1(0(1(0(1(0(1(0(1(1(1(1(0(1(1(1(1(1(0(0(1(1(0(0(1(1(1(1(0(1(0(0(0(1(0(1(1(0(0(1(1(0(0(0(0(0(1(1(0(1(0(1(0(1(1(0(0(0(0(1(0(1(0(0(0(0(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 11.06/3.53 0(0(0(1(0(1(0(1(0(1(1(0(0(1(0(1(0(0(0(1(1(0(1(1(1(1(0(1(1(0(0(0(1(0(0(1(0(0(1(0(0(1(1(1(1(0(1(1(1(1(1(1(1(1(0(0(1(0(0(1(0(0(0(0(1(0(0(0(1(0(0(1(0(1(0(1(0(1(0(0(1(0(0(0(1(1(0(0(0(0(1(1(1(1(0(0(1(1(1(1(0(0(1(1(1(0(0(0(0(1(0(0(1(0(0(0(0(0(1(1(1(0(1(1(1(1(1(1(1(1(1(1(0(0(0(0(1(0(1(0(1(0(1(1(0(0(1(1(0(1(1(1(0(1(1(0(0(1(1(0(1(1(0(0(0(0(1(1(1(0(0(1(1(0(1(0(1(1(1(0(0(1(0(0(1(0(0(1(0(1(1(1(0(0(1(0(0(0(1(0(1(1(1(0(0(1(0(0(1(1(1(1(1(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(0(1(1(0(0(1(0(0(1(0(0(0(0(1(1(0(1(1(0(0(0(1(1(1(1(0(1(0(0(0(0(0(0(0(1(0(1(1(0(1(1(0(1(1(0(1(1(1(0(0(0(1(1(1(1(0(0(1(1(1(0(0(1(0(0(1(1(0(0(0(0(1(1(0(0(0(0(0(1(1(0(1(0(0(1(0(1(0(1(0(0(1(1(1(1(1(0(1(1(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(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 11.06/3.53 11.06/3.53 11.06/3.53 11.06/3.53 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (2) 11.06/3.53 Obligation: 11.06/3.53 Q restricted rewrite system: 11.06/3.53 The TRS R consists of the following rules: 11.06/3.53 11.06/3.53 3(4(5(x1))) -> 4(3(5(x1))) 11.06/3.53 11.06/3.53 Q is empty. 11.06/3.53 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (3) Overlay + Local Confluence (EQUIVALENT) 11.06/3.53 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (4) 11.06/3.53 Obligation: 11.06/3.53 Q restricted rewrite system: 11.06/3.53 The TRS R consists of the following rules: 11.06/3.53 11.06/3.53 3(4(5(x1))) -> 4(3(5(x1))) 11.06/3.53 11.06/3.53 The set Q consists of the following terms: 11.06/3.53 11.06/3.53 3(4(5(x0))) 11.06/3.53 11.06/3.53 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (5) DependencyPairsProof (EQUIVALENT) 11.06/3.53 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (6) 11.06/3.53 Obligation: 11.06/3.53 Q DP problem: 11.06/3.53 The TRS P consists of the following rules: 11.06/3.53 11.06/3.53 3^1(4(5(x1))) -> 3^1(5(x1)) 11.06/3.53 11.06/3.53 The TRS R consists of the following rules: 11.06/3.53 11.06/3.53 3(4(5(x1))) -> 4(3(5(x1))) 11.06/3.53 11.06/3.53 The set Q consists of the following terms: 11.06/3.53 11.06/3.53 3(4(5(x0))) 11.06/3.53 11.06/3.53 We have to consider all minimal (P,Q,R)-chains. 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (7) DependencyGraphProof (EQUIVALENT) 11.06/3.53 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 11.06/3.53 ---------------------------------------- 11.06/3.53 11.06/3.53 (8) 11.06/3.53 TRUE 12.65/3.98 EOF