4.27/1.93 YES 4.51/1.94 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.51/1.94 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.51/1.94 4.51/1.94 4.51/1.94 Termination w.r.t. Q of the given QTRS could be proven: 4.51/1.94 4.51/1.94 (0) QTRS 4.51/1.94 (1) QTRSRRRProof [EQUIVALENT, 43 ms] 4.51/1.94 (2) QTRS 4.51/1.94 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.51/1.94 (4) QDP 4.51/1.94 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.51/1.94 (6) TRUE 4.51/1.94 4.51/1.94 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (0) 4.51/1.94 Obligation: 4.51/1.94 Q restricted rewrite system: 4.51/1.94 The TRS R consists of the following rules: 4.51/1.94 4.51/1.94 f(0) -> f(0) 4.51/1.94 0 -> 1 4.51/1.94 4.51/1.94 The set Q consists of the following terms: 4.51/1.94 4.51/1.94 0 4.51/1.94 4.51/1.94 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (1) QTRSRRRProof (EQUIVALENT) 4.51/1.94 Used ordering: 4.51/1.94 Polynomial interpretation [POLO]: 4.51/1.94 4.51/1.94 POL(0) = 1 4.51/1.94 POL(1) = 0 4.51/1.94 POL(f(x_1)) = x_1 4.51/1.94 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 4.51/1.94 4.51/1.94 0 -> 1 4.51/1.94 4.51/1.94 4.51/1.94 4.51/1.94 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (2) 4.51/1.94 Obligation: 4.51/1.94 Q restricted rewrite system: 4.51/1.94 The TRS R consists of the following rules: 4.51/1.94 4.51/1.94 f(0) -> f(0) 4.51/1.94 4.51/1.94 The set Q consists of the following terms: 4.51/1.94 4.51/1.94 0 4.51/1.94 4.51/1.94 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (3) DependencyPairsProof (EQUIVALENT) 4.51/1.94 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (4) 4.51/1.94 Obligation: 4.51/1.94 Q DP problem: 4.51/1.94 The TRS P consists of the following rules: 4.51/1.94 4.51/1.94 F(0) -> F(0) 4.51/1.94 4.51/1.94 The TRS R consists of the following rules: 4.51/1.94 4.51/1.94 f(0) -> f(0) 4.51/1.94 4.51/1.94 The set Q consists of the following terms: 4.51/1.94 4.51/1.94 0 4.51/1.94 4.51/1.94 We have to consider all minimal (P,Q,R)-chains. 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (5) DependencyGraphProof (EQUIVALENT) 4.51/1.94 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 4.51/1.94 ---------------------------------------- 4.51/1.94 4.51/1.94 (6) 4.51/1.94 TRUE 4.56/1.97 EOF