55.72/15.10 YES 55.98/15.13 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 55.98/15.13 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 55.98/15.13 55.98/15.13 55.98/15.13 Termination w.r.t. Q of the given QTRS could be proven: 55.98/15.13 55.98/15.13 (0) QTRS 55.98/15.13 (1) DependencyPairsProof [EQUIVALENT, 294 ms] 55.98/15.13 (2) QDP 55.98/15.13 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 55.98/15.13 (4) QDP 55.98/15.13 (5) MRRProof [EQUIVALENT, 819 ms] 55.98/15.13 (6) QDP 55.98/15.13 (7) QDPOrderProof [EQUIVALENT, 59 ms] 55.98/15.13 (8) QDP 55.98/15.13 (9) QDPOrderProof [EQUIVALENT, 39 ms] 55.98/15.13 (10) QDP 55.98/15.13 (11) PisEmptyProof [EQUIVALENT, 0 ms] 55.98/15.13 (12) YES 55.98/15.13 55.98/15.13 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (0) 55.98/15.13 Obligation: 55.98/15.13 Q restricted rewrite system: 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (1) DependencyPairsProof (EQUIVALENT) 55.98/15.13 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (2) 55.98/15.13 Obligation: 55.98/15.13 Q DP problem: 55.98/15.13 The TRS P consists of the following rules: 55.98/15.13 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(1(0(2(2(x1))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(2(2(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(1(2(3(2(x1))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(2(x1)) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(2(2(1(3(x1))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(x1) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(3(2(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(2(2(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(3(2(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(0(2(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(2(0(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(2(x1)) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(1(0(4(2(2(x1)))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(4(2(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(2(1(3(2(3(x1)))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(2(3(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(4(2(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 5^1(0(4(2(2(x1))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(3(1(3(2(x1))))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(1(3(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(0(2(0(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(2(0(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(3(2(0(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(2(0(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 5^1(3(0(2(x1)))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(0(2(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 3^1(0(5(x1))) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(0(4(1(x1)))) 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(4(1(x1))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(0(4(1(5(x1))))) 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(4(1(5(x1)))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(4(1(5(0(x1))))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(0(x1)) 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(4(1(5(5(0(x1)))))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(5(0(x1))) 55.98/15.13 5^1(1(2(2(x1)))) -> 0^1(2(2(5(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(5(2(2(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(2(2(x1))) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(2(3(2(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(2(x1)) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(0(2(2(3(x1))))) 55.98/15.13 5^1(1(2(2(x1)))) -> 0^1(2(2(3(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(x1) 55.98/15.13 5^1(1(2(2(x1)))) -> 0^1(3(2(5(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(2(5(x1))) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(1(3(5(2(2(x1)))))) 55.98/15.13 5^1(1(2(2(x1)))) -> 3^1(2(2(5(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(1(0(4(2(2(x1)))))) 55.98/15.13 5^1(1(2(2(x1)))) -> 0^1(4(2(2(x1)))) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(1(2(0(4(2(x1)))))) 55.98/15.13 5^1(1(2(2(x1)))) -> 0^1(4(2(x1))) 55.98/15.13 0^1(1(1(4(5(x1))))) -> 3^1(1(0(4(1(5(x1)))))) 55.98/15.13 0^1(1(1(4(5(x1))))) -> 0^1(4(1(5(x1)))) 55.98/15.13 0^1(1(2(2(2(x1))))) -> 0^1(2(2(5(2(x1))))) 55.98/15.13 0^1(1(2(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 0^1(1(2(2(5(x1))))) -> 5^1(0(4(2(2(x1))))) 55.98/15.13 0^1(1(2(2(5(x1))))) -> 0^1(4(2(2(x1)))) 55.98/15.13 0^1(1(2(4(5(x1))))) -> 5^1(1(0(4(5(x1))))) 55.98/15.13 0^1(1(2(4(5(x1))))) -> 0^1(4(5(x1))) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 0^1(4(2(0(5(x1))))) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 5^1(x1) 55.98/15.13 0^1(1(4(5(5(x1))))) -> 5^1(0(4(0(1(5(x1)))))) 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(4(0(1(5(x1))))) 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(1(5(x1))) 55.98/15.13 0^1(1(5(4(5(x1))))) -> 5^1(0(4(1(5(x1))))) 55.98/15.13 0^1(1(5(4(5(x1))))) -> 0^1(4(1(5(x1)))) 55.98/15.13 0^1(5(1(2(2(x1))))) -> 0^1(1(3(2(5(2(x1)))))) 55.98/15.13 0^1(5(1(2(2(x1))))) -> 3^1(2(5(2(x1)))) 55.98/15.13 0^1(5(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 3^1(3(1(2(2(x1))))) -> 3^1(2(0(3(2(x1))))) 55.98/15.13 3^1(3(1(2(2(x1))))) -> 0^1(3(2(x1))) 55.98/15.13 3^1(3(1(2(2(x1))))) -> 3^1(2(x1)) 55.98/15.13 3^1(4(4(0(5(x1))))) -> 3^1(5(4(5(0(4(x1)))))) 55.98/15.13 3^1(4(4(0(5(x1))))) -> 5^1(4(5(0(4(x1))))) 55.98/15.13 3^1(4(4(0(5(x1))))) -> 5^1(0(4(x1))) 55.98/15.13 3^1(4(4(0(5(x1))))) -> 0^1(4(x1)) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 3^1(2(0(5(2(x1))))) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 0^1(5(2(x1))) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 5^1(1(2(2(5(x1))))) -> 5^1(2(3(2(5(x1))))) 55.98/15.13 5^1(1(2(2(5(x1))))) -> 3^1(2(5(x1))) 55.98/15.13 5^1(2(1(2(2(x1))))) -> 3^1(5(2(2(x1)))) 55.98/15.13 5^1(2(1(2(2(x1))))) -> 5^1(2(2(x1))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 0^1(4(2(5(5(5(x1)))))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(5(x1))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(x1)) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 0^1(4(5(4(2(5(x1)))))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(4(2(5(x1)))) 55.98/15.13 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 We have to consider all minimal (P,Q,R)-chains. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (3) DependencyGraphProof (EQUIVALENT) 55.98/15.13 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 78 less nodes. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (4) 55.98/15.13 Obligation: 55.98/15.13 Q DP problem: 55.98/15.13 The TRS P consists of the following rules: 55.98/15.13 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 0^1(5(2(x1))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(0(x1)) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(5(x1))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(x1)) 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(5(0(x1))) 55.98/15.13 0^1(1(2(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 5^1(x1) 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(1(5(x1))) 55.98/15.13 0^1(5(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 We have to consider all minimal (P,Q,R)-chains. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (5) MRRProof (EQUIVALENT) 55.98/15.13 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 55.98/15.13 55.98/15.13 Strictly oriented dependency pairs: 55.98/15.13 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(2(2(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 5^1(1(2(2(x1)))) -> 5^1(x1) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 0^1(5(2(x1))) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(0(x1)) 55.98/15.13 5^1(0(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(5(x1))) 55.98/15.13 5^1(2(4(0(5(x1))))) -> 5^1(5(x1)) 55.98/15.13 0^1(1(4(5(x1)))) -> 5^1(5(0(x1))) 55.98/15.13 0^1(1(2(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 0^1(5(x1)) 55.98/15.13 0^1(1(4(5(2(x1))))) -> 5^1(x1) 55.98/15.13 0^1(5(1(2(2(x1))))) -> 5^1(2(x1)) 55.98/15.13 55.98/15.13 55.98/15.13 Used ordering: Polynomial interpretation [POLO]: 55.98/15.13 55.98/15.13 POL(0(x_1)) = x_1 55.98/15.13 POL(0^1(x_1)) = 2 + 2*x_1 55.98/15.13 POL(1(x_1)) = x_1 55.98/15.13 POL(2(x_1)) = 2 + x_1 55.98/15.13 POL(3(x_1)) = x_1 55.98/15.13 POL(4(x_1)) = x_1 55.98/15.13 POL(5(x_1)) = x_1 55.98/15.13 POL(5^1(x_1)) = 2*x_1 55.98/15.13 55.98/15.13 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (6) 55.98/15.13 Obligation: 55.98/15.13 Q DP problem: 55.98/15.13 The TRS P consists of the following rules: 55.98/15.13 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(x1) 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(1(5(x1))) 55.98/15.13 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 We have to consider all minimal (P,Q,R)-chains. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (7) QDPOrderProof (EQUIVALENT) 55.98/15.13 We use the reduction pair processor [LPAR04,JAR06]. 55.98/15.13 55.98/15.13 55.98/15.13 The following pairs can be oriented strictly and are deleted. 55.98/15.13 55.98/15.13 0^1(1(4(5(x1)))) -> 0^1(x1) 55.98/15.13 The remaining pairs can at least be oriented weakly. 55.98/15.13 Used ordering: Polynomial interpretation [POLO]: 55.98/15.13 55.98/15.13 POL(0(x_1)) = x_1 55.98/15.13 POL(0^1(x_1)) = x_1 55.98/15.13 POL(1(x_1)) = 1 + x_1 55.98/15.13 POL(2(x_1)) = 0 55.98/15.13 POL(3(x_1)) = 0 55.98/15.13 POL(4(x_1)) = x_1 55.98/15.13 POL(5(x_1)) = x_1 55.98/15.13 55.98/15.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 55.98/15.13 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (8) 55.98/15.13 Obligation: 55.98/15.13 Q DP problem: 55.98/15.13 The TRS P consists of the following rules: 55.98/15.13 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(1(5(x1))) 55.98/15.13 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 We have to consider all minimal (P,Q,R)-chains. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (9) QDPOrderProof (EQUIVALENT) 55.98/15.13 We use the reduction pair processor [LPAR04,JAR06]. 55.98/15.13 55.98/15.13 55.98/15.13 The following pairs can be oriented strictly and are deleted. 55.98/15.13 55.98/15.13 0^1(1(4(5(5(x1))))) -> 0^1(1(5(x1))) 55.98/15.13 The remaining pairs can at least be oriented weakly. 55.98/15.13 Used ordering: Polynomial interpretation [POLO]: 55.98/15.13 55.98/15.13 POL(0(x_1)) = 0 55.98/15.13 POL(0^1(x_1)) = x_1 55.98/15.13 POL(1(x_1)) = x_1 55.98/15.13 POL(2(x_1)) = 0 55.98/15.13 POL(3(x_1)) = 0 55.98/15.13 POL(4(x_1)) = x_1 55.98/15.13 POL(5(x_1)) = 1 + x_1 55.98/15.13 55.98/15.13 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 55.98/15.13 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (10) 55.98/15.13 Obligation: 55.98/15.13 Q DP problem: 55.98/15.13 P is empty. 55.98/15.13 The TRS R consists of the following rules: 55.98/15.13 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(2(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(2(1(3(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(0(3(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(0(3(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(0(2(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(3(2(0(2(x1))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(1(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 0(2(1(3(2(3(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(2(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(0(3(1(3(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(1(0(4(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(0(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(3(3(2(0(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(1(5(3(0(2(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(2(1(3(0(5(x1)))))) 55.98/15.13 0(1(2(2(x1)))) -> 2(4(1(3(2(0(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(5(0(4(1(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(0(4(1(5(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(0(x1))))) 55.98/15.13 0(1(4(5(x1)))) -> 1(1(5(0(4(1(x1)))))) 55.98/15.13 0(1(4(5(x1)))) -> 5(4(1(5(5(0(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(0(2(2(5(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(3(5(2(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(2(3(2(x1))))) 55.98/15.13 5(1(2(2(x1)))) -> 1(5(0(2(2(3(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 2(1(0(3(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 3(1(3(5(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 4(1(3(2(2(5(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(0(4(2(2(x1)))))) 55.98/15.13 5(1(2(2(x1)))) -> 5(1(2(0(4(2(x1)))))) 55.98/15.13 0(1(1(4(5(x1))))) -> 3(1(0(4(1(5(x1)))))) 55.98/15.13 0(1(2(2(2(x1))))) -> 1(0(2(2(5(2(x1)))))) 55.98/15.13 0(1(2(2(5(x1))))) -> 1(5(0(4(2(2(x1)))))) 55.98/15.13 0(1(2(4(5(x1))))) -> 2(5(1(0(4(5(x1)))))) 55.98/15.13 0(1(4(5(2(x1))))) -> 1(0(4(2(0(5(x1)))))) 55.98/15.13 0(1(4(5(5(x1))))) -> 5(0(4(0(1(5(x1)))))) 55.98/15.13 0(1(5(4(5(x1))))) -> 1(5(0(4(1(5(x1)))))) 55.98/15.13 0(5(1(2(2(x1))))) -> 0(1(3(2(5(2(x1)))))) 55.98/15.13 3(3(1(2(2(x1))))) -> 1(3(2(0(3(2(x1)))))) 55.98/15.13 3(4(4(0(5(x1))))) -> 3(5(4(5(0(4(x1)))))) 55.98/15.13 5(0(1(2(2(x1))))) -> 1(3(2(0(5(2(x1)))))) 55.98/15.13 5(1(2(2(5(x1))))) -> 1(5(2(3(2(5(x1)))))) 55.98/15.13 5(2(1(2(2(x1))))) -> 2(1(3(5(2(2(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(2(5(5(5(x1)))))) 55.98/15.13 5(2(4(0(5(x1))))) -> 0(4(5(4(2(5(x1)))))) 55.98/15.13 55.98/15.13 Q is empty. 55.98/15.13 We have to consider all minimal (P,Q,R)-chains. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (11) PisEmptyProof (EQUIVALENT) 55.98/15.13 The TRS P is empty. Hence, there is no (P,Q,R) chain. 55.98/15.13 ---------------------------------------- 55.98/15.13 55.98/15.13 (12) 55.98/15.13 YES 56.14/15.21 EOF