12.08/3.84 YES 12.08/3.88 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 12.08/3.88 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.08/3.88 12.08/3.88 12.08/3.88 Termination w.r.t. Q of the given QTRS could be proven: 12.08/3.88 12.08/3.88 (0) QTRS 12.08/3.88 (1) QTRSRRRProof [EQUIVALENT, 175 ms] 12.08/3.88 (2) QTRS 12.08/3.88 (3) Overlay + Local Confluence [EQUIVALENT, 0 ms] 12.08/3.88 (4) QTRS 12.08/3.88 (5) DependencyPairsProof [EQUIVALENT, 25 ms] 12.08/3.88 (6) QDP 12.08/3.88 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 12.08/3.88 (8) TRUE 12.08/3.88 12.08/3.88 12.08/3.88 ---------------------------------------- 12.08/3.88 12.08/3.88 (0) 12.08/3.88 Obligation: 12.08/3.88 Q restricted rewrite system: 12.08/3.88 The TRS R consists of the following rules: 12.08/3.88 12.08/3.88 0(x1) -> 1(x1) 12.08/3.88 0(0(x1)) -> 0(x1) 12.08/3.88 3(4(5(x1))) -> 4(3(5(x1))) 12.08/3.88 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(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(0(0(1(1(1(0(1(1(1(1(1(0(0(1(0(0(1(0(1(1(0(1(0(0(1(1(0(0(0(1(0(1(1(0(0(0(1(0(0(0(1(1(0(0(0(1(1(1(0(1(1(0(0(1(0(1(1(1(0(1(0(0(0(1(1(0(0(1(1(1(1(1(0(1(0(0(0(1(0(0(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(0(0(0(0(1(1(1(0(1(1(0(1(1(1(0(1(0(0(1(0(0(0(0(0(0(1(0(0(1(0(1(1(0(1(0(1(1(1(1(1(1(1(1(1(0(1(1(1(1(1(1(0(1(1(1(1(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(1(0(1(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(0(0(1(1(1(0(0(0(0(0(1(0(1(1(0(1(1(1(0(0(0(0(1(1(0(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(0(0(1(0(0(0(1(0(0(1(1(0(0(0(0(1(1(1(0(0(0(1(1(1(0(1(1(1(1(1(1(1(1(1(1(1(0(0(0(1(0(0(0(1(1(0(1(0(0(1(1(1(1(0(1(1(1(1(1(0(0(0(0(0(0(1(1(0(0(0(1(0(0(1(1(1(1(0(0(0(0(1(0(0(1(1(0(1(0(1(1(0(0(0(1(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 12.08/3.88 0(1(0(0(1(1(1(1(1(1(0(1(1(0(1(1(0(1(0(0(0(1(0(0(0(1(1(1(1(0(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(0(1(1(0(1(0(0(1(0(1(1(0(1(0(1(1(1(0(1(1(1(0(0(0(0(1(0(0(1(1(0(0(1(1(0(0(1(0(0(1(1(1(1(1(1(1(0(1(0(0(0(1(0(1(0(0(0(1(1(0(1(0(1(1(1(1(0(0(0(0(1(0(1(1(1(1(1(0(1(1(1(0(0(1(0(1(1(1(1(0(1(1(0(0(0(1(0(0(1(0(1(0(1(1(0(0(1(1(0(1(0(0(1(1(1(0(1(0(1(0(1(1(0(1(0(1(0(0(1(0(0(0(0(1(1(1(1(0(1(0(0(0(0(0(0(0(1(1(0(1(1(0(1(0(0(1(1(1(1(0(1(0(1(0(0(0(1(0(1(0(0(1(0(1(0(0(1(1(0(1(1(1(0(1(0(1(0(0(0(1(0(0(1(0(1(0(0(1(1(1(0(1(0(1(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(1(0(1(0(1(0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(1(0(1(0(0(0(0(1(1(1(0(0(0(0(0(1(1(1(0(1(0(1(0(0(0(0(1(1(1(0(0(1(1(1(1(0(1(0(0(1(0(1(0(1(0(1(0(0(1(1(1(1(1(0(0(0(1(0(0(0(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(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 12.08/3.88 12.08/3.88 Q is empty. 12.08/3.88 12.08/3.88 ---------------------------------------- 12.08/3.88 12.08/3.88 (1) QTRSRRRProof (EQUIVALENT) 12.08/3.88 Used ordering: 12.08/3.88 Polynomial interpretation [POLO]: 12.08/3.88 12.08/3.88 POL(0(x_1)) = 14 + x_1 12.08/3.88 POL(1(x_1)) = 13 + x_1 12.08/3.88 POL(2(x_1)) = 70 + x_1 12.08/3.88 POL(3(x_1)) = x_1 12.08/3.88 POL(4(x_1)) = x_1 12.08/3.88 POL(5(x_1)) = x_1 12.08/3.88 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 12.08/3.88 12.08/3.88 0(x1) -> 1(x1) 12.08/3.88 0(0(x1)) -> 0(x1) 12.08/3.88 2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(2(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(0(0(1(1(1(0(1(1(1(1(1(0(0(1(0(0(1(0(1(1(0(1(0(0(1(1(0(0(0(1(0(1(1(0(0(0(1(0(0(0(1(1(0(0(0(1(1(1(0(1(1(0(0(1(0(1(1(1(0(1(0(0(0(1(1(0(0(1(1(1(1(1(0(1(0(0(0(1(0(0(0(0(0(1(0(1(1(0(1(1(0(0(0(1(0(0(0(0(0(1(1(1(0(1(1(0(1(1(1(0(1(0(0(1(0(0(0(0(0(0(1(0(0(1(0(1(1(0(1(0(1(1(1(1(1(1(1(1(1(0(1(1(1(1(1(1(0(1(1(1(1(0(1(0(1(1(1(0(1(1(0(0(1(1(0(1(0(1(0(0(1(0(0(0(0(0(1(0(0(0(1(1(1(0(1(0(1(0(1(1(1(0(0(1(0(1(0(0(0(1(1(0(0(0(1(1(1(0(0(0(0(0(1(0(1(1(0(1(1(1(0(0(0(0(1(1(0(0(0(1(1(1(0(0(0(1(1(0(1(0(0(1(0(0(0(1(0(0(0(1(0(0(1(1(0(0(0(0(1(1(1(0(0(0(1(1(1(0(1(1(1(1(1(1(1(1(1(1(1(0(0(0(1(0(0(0(1(1(0(1(0(0(1(1(1(1(0(1(1(1(1(1(0(0(0(0(0(0(1(1(0(0(0(1(0(0(1(1(1(1(0(0(0(0(1(0(0(1(1(0(1(0(1(1(0(0(0(1(1(0(x1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 12.08/3.88 0(1(0(0(1(1(1(1(1(1(0(1(1(0(1(1(0(1(0(0(0(1(0(0(0(1(1(1(1(0(1(0(1(0(1(1(0(1(1(1(1(1(0(0(1(0(1(1(0(1(0(0(1(0(1(1(0(1(0(1(1(1(0(1(1(1(0(0(0(0(1(0(0(1(1(0(0(1(1(0(0(1(0(0(1(1(1(1(1(1(1(0(1(0(0(0(1(0(1(0(0(0(1(1(0(1(0(1(1(1(1(0(0(0(0(1(0(1(1(1(1(1(0(1(1(1(0(0(1(0(1(1(1(1(0(1(1(0(0(0(1(0(0(1(0(1(0(1(1(0(0(1(1(0(1(0(0(1(1(1(0(1(0(1(0(1(1(0(1(0(1(0(0(1(0(0(0(0(1(1(1(1(0(1(0(0(0(0(0(0(0(1(1(0(1(1(0(1(0(0(1(1(1(1(0(1(0(1(0(0(0(1(0(1(0(0(1(0(1(0(0(1(1(0(1(1(1(0(1(0(1(0(0(0(1(0(0(1(0(1(0(0(1(1(1(0(1(0(1(1(0(1(0(0(0(1(1(1(1(1(0(0(0(0(1(0(1(0(1(0(0(0(1(0(0(0(0(0(1(0(1(1(0(0(0(0(0(1(0(1(0(0(0(0(1(1(1(0(0(0(0(0(1(1(1(0(1(0(1(0(0(0(0(1(1(1(0(0(1(1(1(1(0(1(0(0(1(0(1(0(1(0(1(0(0(1(1(1(1(1(0(0(0(1(0(0(0(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(x1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 12.08/3.89 12.08/3.89 12.08/3.89 12.08/3.89 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (2) 12.08/3.89 Obligation: 12.08/3.89 Q restricted rewrite system: 12.08/3.89 The TRS R consists of the following rules: 12.08/3.89 12.08/3.89 3(4(5(x1))) -> 4(3(5(x1))) 12.08/3.89 12.08/3.89 Q is empty. 12.08/3.89 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (3) Overlay + Local Confluence (EQUIVALENT) 12.08/3.89 The TRS is overlay and locally confluent. By [NOC] we can switch to innermost. 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (4) 12.08/3.89 Obligation: 12.08/3.89 Q restricted rewrite system: 12.08/3.89 The TRS R consists of the following rules: 12.08/3.89 12.08/3.89 3(4(5(x1))) -> 4(3(5(x1))) 12.08/3.89 12.08/3.89 The set Q consists of the following terms: 12.08/3.89 12.08/3.89 3(4(5(x0))) 12.08/3.89 12.08/3.89 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (5) DependencyPairsProof (EQUIVALENT) 12.08/3.89 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (6) 12.08/3.89 Obligation: 12.08/3.89 Q DP problem: 12.08/3.89 The TRS P consists of the following rules: 12.08/3.89 12.08/3.89 3^1(4(5(x1))) -> 3^1(5(x1)) 12.08/3.89 12.08/3.89 The TRS R consists of the following rules: 12.08/3.89 12.08/3.89 3(4(5(x1))) -> 4(3(5(x1))) 12.08/3.89 12.08/3.89 The set Q consists of the following terms: 12.08/3.89 12.08/3.89 3(4(5(x0))) 12.08/3.89 12.08/3.89 We have to consider all minimal (P,Q,R)-chains. 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (7) DependencyGraphProof (EQUIVALENT) 12.08/3.89 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 12.08/3.89 ---------------------------------------- 12.08/3.89 12.08/3.89 (8) 12.08/3.89 TRUE 12.50/3.96 EOF