10.65/3.49 YES 17.65/5.33 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 17.65/5.33 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 17.65/5.33 17.65/5.33 17.65/5.33 Termination w.r.t. Q of the given QTRS could be proven: 17.65/5.33 17.65/5.33 (0) QTRS 17.65/5.33 (1) QTRSRRRProof [EQUIVALENT, 110 ms] 17.65/5.33 (2) QTRS 17.65/5.33 (3) Overlay + Local Confluence [EQUIVALENT, 11 ms] 17.65/5.33 (4) QTRS 17.65/5.33 (5) DependencyPairsProof [EQUIVALENT, 24 ms] 17.65/5.33 (6) QDP 17.65/5.33 (7) DependencyGraphProof [EQUIVALENT, 1 ms] 17.65/5.33 (8) TRUE 17.65/5.33 17.65/5.33 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (0) 17.65/5.33 Obligation: 17.65/5.33 Q restricted rewrite system: 17.65/5.33 The TRS R consists of the following rules: 17.65/5.33 17.65/5.33 0(x1) -> 1(x1) 17.65/5.33 0(0(x1)) -> 0(x1) 17.65/5.33 3(4(5(x1))) -> 4(3(5(x1))) 17.65/5.33 2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))) -> 0(0(0(1(0(1(1(1(0(1(1(1(1(0(0(0(0(1(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(1(1(1(1(1(1(1(0(0(0(1(0(1(1(1(0(0(0(1(1(1(1(1(1(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(0(1(0(1(0(0(1(1(1(0(1(0(0(0(0(1(0(1(1(1(1(0(0(0(0(1(0(0(0(0(0(0(0(0(0(1(1(1(1(0(1(0(1(0(0(0(0(1(1(0(0(1(1(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 17.65/5.33 1(1(0(0(1(1(0(1(0(0(1(0(1(1(1(1(1(0(0(1(0(1(1(1(0(1(0(0(0(0(1(0(1(1(1(1(1(0(0(1(0(0(1(1(0(0(0(0(1(1(0(0(1(0(1(0(0(0(0(0(0(1(0(0(0(1(0(1(1(0(0(1(1(0(1(1(0(1(1(0(1(0(0(1(1(1(0(1(0(0(1(0(0(0(0(0(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(1(1(0(0(1(0(1(1(0(1(0(1(1(1(0(1(0(1(1(0(0(0(1(0(1(1(0(1(1(0(1(1(0(0(0(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))) 17.65/5.33 17.65/5.33 Q is empty. 17.65/5.33 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (1) QTRSRRRProof (EQUIVALENT) 17.65/5.33 Used ordering: 17.65/5.33 Polynomial interpretation [POLO]: 17.65/5.33 17.65/5.33 POL(0(x_1)) = 2 + x_1 17.65/5.33 POL(1(x_1)) = 1 + x_1 17.65/5.33 POL(2(x_1)) = 17 + x_1 17.65/5.33 POL(3(x_1)) = x_1 17.65/5.33 POL(4(x_1)) = x_1 17.65/5.33 POL(5(x_1)) = x_1 17.65/5.33 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 17.65/5.33 17.65/5.33 0(x1) -> 1(x1) 17.65/5.33 0(0(x1)) -> 0(x1) 17.65/5.33 2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))) -> 0(0(0(1(0(1(1(1(0(1(1(1(1(0(0(0(0(1(0(0(1(1(1(1(0(0(0(0(0(0(0(0(1(0(0(0(1(1(1(1(1(1(1(1(1(0(0(0(0(1(1(1(1(1(1(1(0(0(0(1(0(1(1(1(0(0(0(1(1(1(1(1(1(1(1(0(1(0(0(1(0(0(1(1(0(0(0(1(0(1(0(1(0(0(1(1(1(0(1(0(0(0(0(1(0(1(1(1(1(0(0(0(0(1(0(0(0(0(0(0(0(0(0(1(1(1(1(0(1(0(1(0(0(0(0(1(1(0(0(1(1(1(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 17.65/5.33 1(1(0(0(1(1(0(1(0(0(1(0(1(1(1(1(1(0(0(1(0(1(1(1(0(1(0(0(0(0(1(0(1(1(1(1(1(0(0(1(0(0(1(1(0(0(0(0(1(1(0(0(1(0(1(0(0(0(0(0(0(1(0(0(0(1(0(1(1(0(0(1(1(0(1(1(0(1(1(0(1(0(0(1(1(1(0(1(0(0(1(0(0(0(0(0(0(1(1(1(0(1(1(0(0(1(1(0(1(0(0(1(1(0(0(1(0(1(1(0(1(0(1(1(1(0(1(0(1(1(0(0(0(1(0(1(1(0(1(1(0(1(1(0(0(0(1(1(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) -> 2(2(2(2(2(2(2(2(2(2(2(2(2(x1))))))))))))) 17.65/5.33 17.65/5.33 17.65/5.33 17.65/5.33 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (2) 17.65/5.33 Obligation: 17.65/5.33 Q restricted rewrite system: 17.65/5.33 The TRS R consists of the following rules: 17.65/5.33 17.65/5.33 3(4(5(x1))) -> 4(3(5(x1))) 17.65/5.33 17.65/5.33 Q is empty. 17.65/5.33 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (3) Overlay + Local Confluence (EQUIVALENT) 17.65/5.33 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (4) 17.65/5.33 Obligation: 17.65/5.33 Q restricted rewrite system: 17.65/5.33 The TRS R consists of the following rules: 17.65/5.33 17.65/5.33 3(4(5(x1))) -> 4(3(5(x1))) 17.65/5.33 17.65/5.33 The set Q consists of the following terms: 17.65/5.33 17.65/5.33 3(4(5(x0))) 17.65/5.33 17.65/5.33 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (5) DependencyPairsProof (EQUIVALENT) 17.65/5.33 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (6) 17.65/5.33 Obligation: 17.65/5.33 Q DP problem: 17.65/5.33 The TRS P consists of the following rules: 17.65/5.33 17.65/5.33 3^1(4(5(x1))) -> 3^1(5(x1)) 17.65/5.33 17.65/5.33 The TRS R consists of the following rules: 17.65/5.33 17.65/5.33 3(4(5(x1))) -> 4(3(5(x1))) 17.65/5.33 17.65/5.33 The set Q consists of the following terms: 17.65/5.33 17.65/5.33 3(4(5(x0))) 17.65/5.33 17.65/5.33 We have to consider all minimal (P,Q,R)-chains. 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (7) DependencyGraphProof (EQUIVALENT) 17.65/5.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 17.65/5.33 ---------------------------------------- 17.65/5.33 17.65/5.33 (8) 17.65/5.33 TRUE 18.77/7.66 EOF