30.79/8.78 YES 30.79/8.83 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.79/8.83 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.79/8.83 30.79/8.83 30.79/8.83 Termination w.r.t. Q of the given QTRS could be proven: 30.79/8.83 30.79/8.83 (0) QTRS 30.79/8.83 (1) DependencyPairsProof [EQUIVALENT, 653 ms] 30.79/8.83 (2) QDP 30.79/8.83 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 30.79/8.83 (4) AND 30.79/8.83 (5) QDP 30.79/8.83 (6) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.79/8.83 (7) YES 30.79/8.83 (8) QDP 30.79/8.83 (9) UsableRulesProof [EQUIVALENT, 0 ms] 30.79/8.83 (10) QDP 30.79/8.83 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.79/8.83 (12) YES 30.79/8.83 (13) QDP 30.79/8.83 (14) QDPOrderProof [EQUIVALENT, 192 ms] 30.79/8.83 (15) QDP 30.79/8.83 (16) PisEmptyProof [EQUIVALENT, 0 ms] 30.79/8.83 (17) YES 30.79/8.83 30.79/8.83 30.79/8.83 ---------------------------------------- 30.79/8.83 30.79/8.83 (0) 30.79/8.83 Obligation: 30.79/8.83 Q restricted rewrite system: 30.79/8.83 The TRS R consists of the following rules: 30.79/8.83 30.79/8.83 0(1(0(x1))) -> 0(2(1(0(x1)))) 30.79/8.83 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 30.79/8.83 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 30.79/8.83 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 30.79/8.83 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 30.79/8.83 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 30.79/8.83 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 30.79/8.83 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 30.79/8.83 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 30.79/8.83 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 30.79/8.83 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 30.79/8.83 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 30.79/8.83 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 30.79/8.83 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 30.79/8.83 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 30.79/8.83 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 30.79/8.83 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 30.79/8.83 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 30.79/8.83 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 30.79/8.83 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 30.79/8.83 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 30.79/8.83 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 30.79/8.83 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 30.79/8.83 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 30.79/8.83 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 30.79/8.83 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 30.79/8.83 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 30.79/8.83 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 30.79/8.83 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 30.79/8.83 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 30.79/8.83 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 30.79/8.83 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 30.79/8.83 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 30.79/8.83 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 30.79/8.83 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 30.79/8.83 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 30.79/8.83 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 30.79/8.83 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 30.79/8.83 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 30.79/8.83 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 30.79/8.83 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 30.79/8.83 30.79/8.83 Q is empty. 30.79/8.83 30.79/8.83 ---------------------------------------- 30.79/8.83 30.79/8.83 (1) DependencyPairsProof (EQUIVALENT) 30.79/8.83 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.79/8.83 ---------------------------------------- 30.79/8.83 30.79/8.83 (2) 30.79/8.83 Obligation: 30.79/8.83 Q DP problem: 30.79/8.83 The TRS P consists of the following rules: 30.79/8.83 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(1(0(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(2(1(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(2(1(2(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(1(2(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(1(0(2(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(x1)) 30.79/8.83 0^1(1(0(x1))) -> 0^1(3(2(1(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 3^1(2(1(0(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(0(2(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(2(x1))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(2(0(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(0(x1))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(2(0(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(0(x1))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(2(1(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(1(x1))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(0(0(4(1(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(4(1(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(1(x1))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(1(0(4(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(2(1(0(3(2(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(3(2(x1))) 30.79/8.83 0^1(1(0(x1))) -> 3^1(2(x1)) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(0(4(1(3(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(1(3(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 3^1(x1) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(2(2(1(0(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(5(2(1(2(0(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(2(1(2(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(5(2(5(1(0(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(2(5(1(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(1(0(x1))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(5(4(4(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(5(4(4(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(4(4(x1))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(4(4(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(0(0(4(2(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(4(2(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(2(x1))) 30.79/8.83 0^1(1(0(x1))) -> 3^1(0(0(4(1(4(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(0(4(1(4(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 0^1(4(1(4(x1)))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(1(0(2(0(x1))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(5(1(0(0(2(x1)))))) 30.79/8.83 0^1(1(0(x1))) -> 5^1(1(0(0(2(x1))))) 30.79/8.83 0^1(0(1(0(x1)))) -> 0^1(0(2(0(x1)))) 30.79/8.83 0^1(0(1(0(x1)))) -> 0^1(2(0(x1))) 30.79/8.83 0^1(0(1(0(x1)))) -> 0^1(1(5(0(0(2(x1)))))) 30.79/8.83 0^1(0(1(0(x1)))) -> 5^1(0(0(2(x1)))) 30.79/8.83 0^1(0(1(0(x1)))) -> 0^1(0(2(x1))) 30.79/8.83 0^1(0(1(0(x1)))) -> 0^1(2(x1)) 30.79/8.83 0^1(1(0(3(x1)))) -> 0^1(3(3(0(2(x1))))) 30.79/8.83 0^1(1(0(3(x1)))) -> 3^1(3(0(2(x1)))) 30.79/8.83 0^1(1(0(3(x1)))) -> 3^1(0(2(x1))) 30.79/8.83 0^1(1(0(3(x1)))) -> 0^1(2(x1)) 30.79/8.83 0^1(1(0(3(x1)))) -> 0^1(5(3(2(0(x1))))) 30.79/8.83 0^1(1(0(3(x1)))) -> 5^1(3(2(0(x1)))) 30.79/8.83 0^1(1(0(3(x1)))) -> 3^1(2(0(x1))) 30.79/8.83 0^1(1(0(3(x1)))) -> 0^1(x1) 30.79/8.83 0^1(1(1(0(x1)))) -> 0^1(4(4(1(1(0(x1)))))) 30.79/8.83 0^1(1(1(3(x1)))) -> 3^1(4(5(1(1(0(x1)))))) 30.79/8.83 0^1(1(1(3(x1)))) -> 5^1(1(1(0(x1)))) 30.79/8.83 0^1(1(1(3(x1)))) -> 0^1(x1) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(2(0(x1))) 30.79/8.83 0^1(1(2(0(x1)))) -> 3^1(0(2(1(0(x1))))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(2(1(0(x1)))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(0(2(x1))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(2(x1)) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(0(4(2(5(1(x1)))))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(4(2(5(1(x1))))) 30.79/8.83 0^1(1(2(0(x1)))) -> 5^1(1(x1)) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(4(0(x1))) 30.79/8.83 0^1(1(2(0(x1)))) -> 3^1(0(2(1(0(4(x1)))))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(2(1(0(4(x1))))) 30.79/8.83 0^1(1(2(0(x1)))) -> 0^1(4(x1)) 30.79/8.83 0^1(1(3(0(x1)))) -> 0^1(3(0(2(x1)))) 30.79/8.83 0^1(1(3(0(x1)))) -> 3^1(0(2(x1))) 30.79/8.83 0^1(1(3(0(x1)))) -> 0^1(2(x1)) 30.79/8.83 0^1(1(4(0(x1)))) -> 0^1(3(4(2(1(0(x1)))))) 30.79/8.83 0^1(1(4(0(x1)))) -> 3^1(4(2(1(0(x1))))) 30.79/8.83 0^1(1(5(0(x1)))) -> 0^1(5(1(4(0(x1))))) 30.79/8.83 0^1(1(5(0(x1)))) -> 5^1(1(4(0(x1)))) 30.79/8.83 0^1(1(5(0(x1)))) -> 5^1(3(0(2(0(x1))))) 30.79/8.83 0^1(1(5(0(x1)))) -> 3^1(0(2(0(x1)))) 30.79/8.83 0^1(1(5(0(x1)))) -> 0^1(2(0(x1))) 30.79/8.83 0^1(3(1(0(x1)))) -> 3^1(0(5(0(x1)))) 30.79/8.83 0^1(3(1(0(x1)))) -> 0^1(5(0(x1))) 30.79/8.83 0^1(3(1(0(x1)))) -> 5^1(0(x1)) 30.79/8.83 5^1(0(1(0(x1)))) -> 0^1(0(5(1(x1)))) 30.79/8.83 5^1(0(1(0(x1)))) -> 0^1(5(1(x1))) 30.79/8.83 5^1(0(1(0(x1)))) -> 5^1(1(x1)) 30.79/8.83 5^1(0(1(0(x1)))) -> 0^1(0(4(5(x1)))) 30.79/8.83 5^1(0(1(0(x1)))) -> 0^1(4(5(x1))) 30.79/8.83 5^1(0(1(0(x1)))) -> 5^1(x1) 30.79/8.83 0^1(1(0(0(0(x1))))) -> 0^1(0(5(1(0(0(x1)))))) 30.79/8.83 0^1(1(0(0(0(x1))))) -> 0^1(5(1(0(0(x1))))) 30.79/8.83 0^1(1(0(0(0(x1))))) -> 5^1(1(0(0(x1)))) 30.79/8.83 0^1(1(2(4(0(x1))))) -> 0^1(0(5(4(2(1(x1)))))) 30.79/8.83 0^1(1(2(4(0(x1))))) -> 0^1(5(4(2(1(x1))))) 30.79/8.83 0^1(1(2(4(0(x1))))) -> 5^1(4(2(1(x1)))) 30.79/8.83 0^1(1(2(5(0(x1))))) -> 0^1(2(0(5(4(x1))))) 30.79/8.83 0^1(1(2(5(0(x1))))) -> 0^1(5(4(x1))) 30.79/8.83 0^1(1(2(5(0(x1))))) -> 5^1(4(x1)) 30.79/8.83 0^1(1(4(0(0(x1))))) -> 0^1(0(0(4(1(0(x1)))))) 30.79/8.83 0^1(1(4(0(0(x1))))) -> 0^1(0(4(1(0(x1))))) 30.79/8.83 0^1(1(4(0(0(x1))))) -> 0^1(4(1(0(x1)))) 30.79/8.83 0^1(1(4(5(0(x1))))) -> 5^1(0(0(4(2(x1))))) 30.79/8.83 0^1(1(4(5(0(x1))))) -> 0^1(0(4(2(x1)))) 30.79/8.83 0^1(1(4(5(0(x1))))) -> 0^1(4(2(x1))) 30.79/8.83 0^1(3(0(1(0(x1))))) -> 0^1(3(0(0(2(1(x1)))))) 30.79/8.83 0^1(3(0(1(0(x1))))) -> 3^1(0(0(2(1(x1))))) 30.79/8.83 0^1(3(0(1(0(x1))))) -> 0^1(0(2(1(x1)))) 30.79/8.83 0^1(3(0(1(0(x1))))) -> 0^1(2(1(x1))) 30.79/8.83 3^1(0(3(1(0(x1))))) -> 0^1(1(3(2(3(0(x1)))))) 30.79/8.83 3^1(0(3(1(0(x1))))) -> 3^1(2(3(0(x1)))) 30.79/8.83 3^1(0(3(1(0(x1))))) -> 3^1(0(x1)) 30.79/8.83 5^1(0(1(2(0(x1))))) -> 0^1(0(5(2(1(0(x1)))))) 30.79/8.83 5^1(0(1(2(0(x1))))) -> 0^1(5(2(1(0(x1))))) 30.79/8.83 5^1(0(1(2(0(x1))))) -> 5^1(2(1(0(x1)))) 30.79/8.83 30.79/8.83 The TRS R consists of the following rules: 30.79/8.83 30.79/8.83 0(1(0(x1))) -> 0(2(1(0(x1)))) 30.79/8.83 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 30.79/8.83 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 31.17/8.85 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 31.17/8.85 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 31.17/8.85 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 31.17/8.85 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 31.17/8.85 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 31.17/8.85 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 31.17/8.85 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 31.17/8.85 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 31.17/8.85 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 31.17/8.85 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 31.17/8.85 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 31.17/8.85 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 31.17/8.85 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (3) DependencyGraphProof (EQUIVALENT) 31.17/8.85 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 114 less nodes. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (4) 31.17/8.85 Complex Obligation (AND) 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (5) 31.17/8.85 Obligation: 31.17/8.85 Q DP problem: 31.17/8.85 The TRS P consists of the following rules: 31.17/8.85 31.17/8.85 3^1(0(3(1(0(x1))))) -> 3^1(0(x1)) 31.17/8.85 31.17/8.85 The TRS R consists of the following rules: 31.17/8.85 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(x1)))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 31.17/8.85 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 31.17/8.85 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 31.17/8.85 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 31.17/8.85 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 31.17/8.85 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 31.17/8.85 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 31.17/8.85 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 31.17/8.85 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 31.17/8.85 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 31.17/8.85 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 31.17/8.85 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 31.17/8.85 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 31.17/8.85 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (6) QDPSizeChangeProof (EQUIVALENT) 31.17/8.85 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 31.17/8.85 31.17/8.85 From the DPs we obtained the following set of size-change graphs: 31.17/8.85 *3^1(0(3(1(0(x1))))) -> 3^1(0(x1)) 31.17/8.85 The graph contains the following edges 1 > 1 31.17/8.85 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (7) 31.17/8.85 YES 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (8) 31.17/8.85 Obligation: 31.17/8.85 Q DP problem: 31.17/8.85 The TRS P consists of the following rules: 31.17/8.85 31.17/8.85 5^1(0(1(0(x1)))) -> 5^1(x1) 31.17/8.85 31.17/8.85 The TRS R consists of the following rules: 31.17/8.85 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(x1)))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 31.17/8.85 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 31.17/8.85 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 31.17/8.85 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 31.17/8.85 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 31.17/8.85 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 31.17/8.85 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 31.17/8.85 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 31.17/8.85 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 31.17/8.85 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 31.17/8.85 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 31.17/8.85 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 31.17/8.85 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 31.17/8.85 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (9) UsableRulesProof (EQUIVALENT) 31.17/8.85 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (10) 31.17/8.85 Obligation: 31.17/8.85 Q DP problem: 31.17/8.85 The TRS P consists of the following rules: 31.17/8.85 31.17/8.85 5^1(0(1(0(x1)))) -> 5^1(x1) 31.17/8.85 31.17/8.85 R is empty. 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (11) QDPSizeChangeProof (EQUIVALENT) 31.17/8.85 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 31.17/8.85 31.17/8.85 From the DPs we obtained the following set of size-change graphs: 31.17/8.85 *5^1(0(1(0(x1)))) -> 5^1(x1) 31.17/8.85 The graph contains the following edges 1 > 1 31.17/8.85 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (12) 31.17/8.85 YES 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (13) 31.17/8.85 Obligation: 31.17/8.85 Q DP problem: 31.17/8.85 The TRS P consists of the following rules: 31.17/8.85 31.17/8.85 0^1(1(1(3(x1)))) -> 0^1(x1) 31.17/8.85 0^1(1(0(3(x1)))) -> 0^1(x1) 31.17/8.85 0^1(3(1(0(x1)))) -> 0^1(5(0(x1))) 31.17/8.85 31.17/8.85 The TRS R consists of the following rules: 31.17/8.85 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(x1)))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 31.17/8.85 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 31.17/8.85 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 31.17/8.85 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 31.17/8.85 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 31.17/8.85 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 31.17/8.85 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 31.17/8.85 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 31.17/8.85 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 31.17/8.85 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 31.17/8.85 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 31.17/8.85 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 31.17/8.85 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 31.17/8.85 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (14) QDPOrderProof (EQUIVALENT) 31.17/8.85 We use the reduction pair processor [LPAR04,JAR06]. 31.17/8.85 31.17/8.85 31.17/8.85 The following pairs can be oriented strictly and are deleted. 31.17/8.85 31.17/8.85 0^1(1(1(3(x1)))) -> 0^1(x1) 31.17/8.85 0^1(1(0(3(x1)))) -> 0^1(x1) 31.17/8.85 0^1(3(1(0(x1)))) -> 0^1(5(0(x1))) 31.17/8.85 The remaining pairs can at least be oriented weakly. 31.17/8.85 Used ordering: Polynomial interpretation [POLO]: 31.17/8.85 31.17/8.85 POL(0(x_1)) = x_1 31.17/8.85 POL(0^1(x_1)) = x_1 31.17/8.85 POL(1(x_1)) = x_1 31.17/8.85 POL(2(x_1)) = 0 31.17/8.85 POL(3(x_1)) = 1 + x_1 31.17/8.85 POL(4(x_1)) = 0 31.17/8.85 POL(5(x_1)) = 0 31.17/8.85 31.17/8.85 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 31.17/8.85 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (15) 31.17/8.85 Obligation: 31.17/8.85 Q DP problem: 31.17/8.85 P is empty. 31.17/8.85 The TRS R consists of the following rules: 31.17/8.85 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(x1)))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(3(2(1(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(0(2(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(2(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 1(4(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 4(0(0(2(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(0(0(4(1(x1))))) 31.17/8.85 0(1(0(x1))) -> 5(1(0(4(0(x1))))) 31.17/8.85 0(1(0(x1))) -> 0(2(1(0(3(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(0(4(1(3(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(4(2(2(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(1(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 0(5(2(5(1(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(0(5(4(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(0(4(4(4(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(1(0(x1))) -> 3(0(0(4(1(4(x1)))))) 31.17/8.85 0(1(0(x1))) -> 4(5(1(0(2(0(x1)))))) 31.17/8.85 0(1(0(x1))) -> 5(5(1(0(0(2(x1)))))) 31.17/8.85 0(0(1(0(x1)))) -> 1(0(0(2(0(x1))))) 31.17/8.85 0(0(1(0(x1)))) -> 0(1(5(0(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(3(3(0(2(x1)))))) 31.17/8.85 0(1(0(3(x1)))) -> 1(0(5(3(2(0(x1)))))) 31.17/8.85 0(1(1(0(x1)))) -> 0(4(4(1(1(0(x1)))))) 31.17/8.85 0(1(1(3(x1)))) -> 3(4(5(1(1(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(0(2(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 4(1(0(0(2(x1))))) 31.17/8.85 0(1(2(0(x1)))) -> 0(0(4(2(5(1(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 1(1(2(0(4(0(x1)))))) 31.17/8.85 0(1(2(0(x1)))) -> 3(0(2(1(0(4(x1)))))) 31.17/8.85 0(1(3(0(x1)))) -> 1(0(3(0(2(x1))))) 31.17/8.85 0(1(4(0(x1)))) -> 0(3(4(2(1(0(x1)))))) 31.17/8.85 0(1(5(0(x1)))) -> 0(5(1(4(0(x1))))) 31.17/8.85 0(1(5(0(x1)))) -> 1(5(3(0(2(0(x1)))))) 31.17/8.85 0(3(1(0(x1)))) -> 1(2(3(0(5(0(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 1(4(0(0(5(1(x1)))))) 31.17/8.85 5(0(1(0(x1)))) -> 2(1(0(0(4(5(x1)))))) 31.17/8.85 0(1(0(0(0(x1))))) -> 0(0(5(1(0(0(x1)))))) 31.17/8.85 0(1(2(4(0(x1))))) -> 0(0(5(4(2(1(x1)))))) 31.17/8.85 0(1(2(5(0(x1))))) -> 1(0(2(0(5(4(x1)))))) 31.17/8.85 0(1(4(0(0(x1))))) -> 0(0(0(4(1(0(x1)))))) 31.17/8.85 0(1(4(5(0(x1))))) -> 1(5(0(0(4(2(x1)))))) 31.17/8.85 0(3(0(1(0(x1))))) -> 0(3(0(0(2(1(x1)))))) 31.17/8.85 3(0(3(1(0(x1))))) -> 0(1(3(2(3(0(x1)))))) 31.17/8.85 5(0(1(2(0(x1))))) -> 0(0(5(2(1(0(x1)))))) 31.17/8.85 31.17/8.85 Q is empty. 31.17/8.85 We have to consider all minimal (P,Q,R)-chains. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (16) PisEmptyProof (EQUIVALENT) 31.17/8.85 The TRS P is empty. Hence, there is no (P,Q,R) chain. 31.17/8.85 ---------------------------------------- 31.17/8.85 31.17/8.85 (17) 31.17/8.85 YES 31.30/8.92 EOF