484.46/124.09 YES 486.78/124.66 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 486.78/124.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 486.78/124.66 486.78/124.66 486.78/124.66 Termination w.r.t. Q of the given QTRS could be proven: 486.78/124.66 486.78/124.66 (0) QTRS 486.78/124.66 (1) QTRS Reverse [EQUIVALENT, 0 ms] 486.78/124.66 (2) QTRS 486.78/124.66 (3) DependencyPairsProof [EQUIVALENT, 301 ms] 486.78/124.66 (4) QDP 486.78/124.66 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 486.78/124.66 (6) QDP 486.78/124.66 (7) QDPOrderProof [EQUIVALENT, 8148 ms] 486.78/124.66 (8) QDP 486.78/124.66 (9) QDPOrderProof [EQUIVALENT, 4096 ms] 486.78/124.66 (10) QDP 486.78/124.66 (11) PisEmptyProof [EQUIVALENT, 0 ms] 486.78/124.66 (12) YES 486.78/124.66 486.78/124.66 486.78/124.66 ---------------------------------------- 486.78/124.66 486.78/124.66 (0) 486.78/124.66 Obligation: 486.78/124.66 Q restricted rewrite system: 486.78/124.66 The TRS R consists of the following rules: 486.78/124.66 486.78/124.66 0(1(0(2(x1)))) -> 2(0(3(1(0(x1))))) 486.78/124.66 0(1(0(2(x1)))) -> 2(0(0(3(1(2(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 2(2(0(3(1(0(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 2(3(1(0(0(2(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 2(3(1(0(3(0(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 4(1(0(3(0(2(x1)))))) 486.78/124.66 0(1(0(2(x1)))) -> 4(1(0(4(0(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(3(1(0(4(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(4(0(3(1(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(1(0(4(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(1(4(0(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(0(3(1(2(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(3(2(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(4(2(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(5(2(x1))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(0(3(1(0(4(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(0(3(1(4(4(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(3(1(4(0(4(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(4(3(0(4(1(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 2(4(3(1(0(3(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(1(0(4(1(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(2(1(4(0(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(3(1(0(4(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 3(2(3(1(4(0(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(0(3(1(3(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(0(3(1(4(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(4(3(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(4(5(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(0(5(3(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(1(0(5(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(1(3(0(5(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(3(0(3(1(2(x1)))))) 486.78/124.66 0(1(4(2(x1)))) -> 4(4(0(3(1(2(x1)))))) 486.78/124.66 0(0(1(0(2(x1))))) -> 1(0(0(2(0(4(x1)))))) 486.78/124.66 0(0(1(0(2(x1))))) -> 1(0(4(0(0(2(x1)))))) 486.78/124.66 0(0(1(0(2(x1))))) -> 2(1(0(3(0(0(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 0(0(3(1(2(4(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 0(2(3(1(0(4(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 0(2(4(0(3(1(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 0(3(1(0(2(4(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 1(0(3(4(0(2(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 2(0(0(3(1(4(x1)))))) 486.78/124.66 0(0(1(4(2(x1))))) -> 2(1(0(4(0(0(x1)))))) 486.78/124.66 0(1(2(0(2(x1))))) -> 2(0(1(0(4(2(x1)))))) 486.78/124.66 0(1(2(4(2(x1))))) -> 2(3(1(0(2(4(x1)))))) 486.78/124.66 0(1(2(4(2(x1))))) -> 4(1(0(2(2(4(x1)))))) 486.78/124.66 0(1(3(4(2(x1))))) -> 2(3(1(4(4(0(x1)))))) 486.78/124.66 0(1(3(4(2(x1))))) -> 2(4(3(0(4(1(x1)))))) 486.78/124.66 0(1(3(4(2(x1))))) -> 3(2(1(0(4(0(x1)))))) 486.78/124.66 0(1(3(4(2(x1))))) -> 4(0(3(3(1(2(x1)))))) 486.78/124.66 0(1(3(4(2(x1))))) -> 4(1(4(0(3(2(x1)))))) 486.78/124.66 0(1(4(0(2(x1))))) -> 2(0(3(1(0(4(x1)))))) 486.78/124.66 0(1(5(0(2(x1))))) -> 0(2(3(1(0(5(x1)))))) 486.78/124.66 0(1(5(0(2(x1))))) -> 3(0(5(1(0(2(x1)))))) 486.78/124.66 0(1(5(0(2(x1))))) -> 5(1(3(0(0(2(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 0(4(4(1(2(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 1(0(4(5(1(2(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 2(0(4(4(5(1(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(0(2(3(1(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(1(0(2(5(2(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(1(0(5(2(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(2(1(3(0(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(3(1(0(2(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(4(0(5(1(2(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 4(4(2(1(0(5(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 5(0(4(5(2(1(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 5(1(2(0(4(3(x1)))))) 486.78/124.66 0(1(5(4(2(x1))))) -> 5(3(1(0(4(2(x1)))))) 486.78/124.66 0(2(1(4(2(x1))))) -> 0(4(4(1(2(2(x1)))))) 486.78/124.66 0(2(1(4(2(x1))))) -> 3(2(2(1(4(0(x1)))))) 486.78/124.66 0(2(1(4(2(x1))))) -> 4(1(0(3(2(2(x1)))))) 486.78/124.66 5(0(1(4(2(x1))))) -> 2(0(4(3(5(1(x1)))))) 486.78/124.66 5(0(1(4(2(x1))))) -> 2(4(0(3(1(5(x1)))))) 486.78/124.66 5(0(1(4(2(x1))))) -> 4(1(0(5(3(2(x1)))))) 486.78/124.66 5(0(1(4(2(x1))))) -> 5(2(1(1(0(4(x1)))))) 486.78/124.66 5(0(2(0(2(x1))))) -> 5(0(3(0(2(2(x1)))))) 486.78/124.66 5(0(2(0(2(x1))))) -> 5(0(4(0(2(2(x1)))))) 486.78/124.66 5(0(2(4(2(x1))))) -> 5(4(0(3(2(2(x1)))))) 486.78/124.66 5(1(5(0(2(x1))))) -> 5(2(1(4(5(0(x1)))))) 486.78/124.66 5(1(5(4(2(x1))))) -> 5(2(1(0(4(5(x1)))))) 486.78/124.66 5(4(1(4(2(x1))))) -> 3(2(1(4(4(5(x1)))))) 486.78/124.66 5(4(1(4(2(x1))))) -> 4(4(3(5(1(2(x1)))))) 486.78/124.66 5(4(2(0(2(x1))))) -> 3(0(5(2(2(4(x1)))))) 486.78/124.66 5(4(2(0(2(x1))))) -> 4(0(5(3(2(2(x1)))))) 486.78/124.66 5(4(2(0(2(x1))))) -> 5(2(2(2(4(0(x1)))))) 486.78/124.66 5(4(2(0(2(x1))))) -> 5(3(2(2(4(0(x1)))))) 486.78/124.66 5(4(2(0(2(x1))))) -> 5(4(2(2(4(0(x1)))))) 486.78/124.66 5(4(2(4(2(x1))))) -> 0(4(4(5(2(2(x1)))))) 486.78/124.66 5(4(2(4(2(x1))))) -> 5(4(4(3(2(2(x1)))))) 486.78/124.66 5(4(5(4(2(x1))))) -> 4(5(0(4(5(2(x1)))))) 486.78/124.66 486.78/124.66 Q is empty. 486.78/124.66 486.78/124.66 ---------------------------------------- 486.78/124.66 486.78/124.66 (1) QTRS Reverse (EQUIVALENT) 486.78/124.66 We applied the QTRS Reverse Processor [REVERSE]. 486.78/124.66 ---------------------------------------- 486.78/124.66 486.78/124.66 (2) 486.78/124.66 Obligation: 486.78/124.66 Q restricted rewrite system: 486.78/124.66 The TRS R consists of the following rules: 486.78/124.66 486.78/124.66 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.78/124.66 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.78/124.66 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.78/124.66 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.78/124.66 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.78/124.66 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.78/124.66 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.78/124.66 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.78/124.66 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.78/124.66 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.78/124.66 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.78/124.66 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.78/124.66 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.78/124.66 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.78/124.66 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.78/124.66 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.78/124.66 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.78/124.66 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.78/124.66 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.78/124.66 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.78/124.66 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.78/124.66 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.78/124.66 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.78/124.66 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.78/124.66 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.78/124.66 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.78/124.66 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.78/124.66 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 Q is empty. 486.87/124.70 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (3) DependencyPairsProof (EQUIVALENT) 486.87/124.70 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (4) 486.87/124.70 Obligation: 486.87/124.70 Q DP problem: 486.87/124.70 The TRS P consists of the following rules: 486.87/124.70 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(1(3(0(0(2(x1)))))) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(0(0(1(3(2(x1)))))) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(0(3(0(1(4(x1)))))) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(0(4(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(3(x1)) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(1(3(0(4(x1))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(3(0(1(4(x1))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(4(0(1(4(x1))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(5(0(1(4(x1))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(2(3(x1))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(3(1(3(0(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(4(1(3(0(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(3(4(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(5(4(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(3(5(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(5(0(1(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(5(0(3(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(1(3(0(3(4(x1)))))) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(1(3(0(4(4(x1)))))) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(0(0(1(x1)))) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(0(0(4(0(1(x1)))))) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(1(3(0(0(x1))))) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(1(3(0(x1))))) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(4(3(0(1(x1)))))) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(2(1(0(x1))))) -> 2^1(4(0(1(0(2(x1)))))) 486.87/124.70 2^1(0(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(0(1(3(2(x1))))) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(2(0(1(4(x1))))) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(0(1(4(x1)))) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(3(x1)) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(1(3(3(0(4(x1)))))) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(3(0(4(1(4(x1)))))) 486.87/124.70 2^1(0(4(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(1(5(0(3(x1)))))) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(0(3(1(5(x1)))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(1(4(4(0(x1))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(1(5(4(0(1(x1)))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(5(2(0(1(4(x1)))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(1(4(x1)))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(5(0(1(4(x1))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(1(3(4(x1))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(1(5(0(4(4(x1)))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(4(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(5(4(0(5(x1))))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(1(5(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(0(1(3(5(x1)))))) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(2(1(4(4(0(x1)))))) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(1(4(4(0(x1))))) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(2(3(x1))) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(3(x1)) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(2(3(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(2(0(x1))))) -> 2^1(3(0(1(4(x1))))) 486.87/124.70 2^1(4(1(0(5(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(5(x1))))) -> 2^1(3(5(0(1(4(x1)))))) 486.87/124.70 2^1(4(1(0(5(x1))))) -> 2^1(5(x1)) 486.87/124.70 2^1(0(2(0(5(x1))))) -> 2^1(2(0(3(0(5(x1)))))) 486.87/124.70 2^1(0(2(0(5(x1))))) -> 2^1(0(3(0(5(x1))))) 486.87/124.70 2^1(0(2(0(5(x1))))) -> 2^1(2(0(4(0(5(x1)))))) 486.87/124.70 2^1(0(2(0(5(x1))))) -> 2^1(0(4(0(5(x1))))) 486.87/124.70 2^1(4(2(0(5(x1))))) -> 2^1(2(3(0(4(5(x1)))))) 486.87/124.70 2^1(4(2(0(5(x1))))) -> 2^1(3(0(4(5(x1))))) 486.87/124.70 2^1(0(5(1(5(x1))))) -> 2^1(5(x1)) 486.87/124.70 2^1(4(5(1(5(x1))))) -> 2^1(5(x1)) 486.87/124.70 2^1(4(1(4(5(x1))))) -> 2^1(3(x1)) 486.87/124.70 2^1(4(1(4(5(x1))))) -> 2^1(1(5(3(4(4(x1)))))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(5(0(3(x1))))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(5(0(3(x1)))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(3(5(0(4(x1)))))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(3(5(0(4(x1))))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(2(5(x1)))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(5(x1))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(5(x1)) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(3(5(x1)))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(3(5(x1))) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 486.87/124.70 2^1(4(2(4(5(x1))))) -> 2^1(2(5(4(4(0(x1)))))) 486.87/124.70 2^1(4(2(4(5(x1))))) -> 2^1(5(4(4(0(x1))))) 486.87/124.70 2^1(4(2(4(5(x1))))) -> 2^1(2(3(4(4(5(x1)))))) 486.87/124.70 2^1(4(2(4(5(x1))))) -> 2^1(3(4(4(5(x1))))) 486.87/124.70 2^1(4(5(4(5(x1))))) -> 2^1(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 The TRS R consists of the following rules: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 Q is empty. 486.87/124.70 We have to consider all minimal (P,Q,R)-chains. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (5) DependencyGraphProof (EQUIVALENT) 486.87/124.70 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 76 less nodes. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (6) 486.87/124.70 Obligation: 486.87/124.70 Q DP problem: 486.87/124.70 The TRS P consists of the following rules: 486.87/124.70 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(4(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 486.87/124.70 2^1(4(1(0(5(x1))))) -> 2^1(x1) 486.87/124.70 486.87/124.70 The TRS R consists of the following rules: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 Q is empty. 486.87/124.70 We have to consider all minimal (P,Q,R)-chains. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (7) QDPOrderProof (EQUIVALENT) 486.87/124.70 We use the reduction pair processor [LPAR04,JAR06]. 486.87/124.70 486.87/124.70 486.87/124.70 The following pairs can be oriented strictly and are deleted. 486.87/124.70 486.87/124.70 2^1(4(1(0(5(x1))))) -> 2^1(x1) 486.87/124.70 The remaining pairs can at least be oriented weakly. 486.87/124.70 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(2^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(0(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(1(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(2(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(4(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, 0A, 0A], [0A, -I, 0A]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, -I, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(5(x_1)) = [[1A], [0A], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 486.87/124.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (8) 486.87/124.70 Obligation: 486.87/124.70 Q DP problem: 486.87/124.70 The TRS P consists of the following rules: 486.87/124.70 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(4(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 486.87/124.70 486.87/124.70 The TRS R consists of the following rules: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 Q is empty. 486.87/124.70 We have to consider all minimal (P,Q,R)-chains. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (9) QDPOrderProof (EQUIVALENT) 486.87/124.70 We use the reduction pair processor [LPAR04,JAR06]. 486.87/124.70 486.87/124.70 486.87/124.70 The following pairs can be oriented strictly and are deleted. 486.87/124.70 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(2(x1)) 486.87/124.70 2^1(0(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(x1)))) -> 2^1(x1) 486.87/124.70 2^1(0(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(1(0(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(2(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(3(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(4(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(0(5(1(0(x1))))) -> 2^1(0(x1)) 486.87/124.70 2^1(0(2(4(5(x1))))) -> 2^1(2(4(5(x1)))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(x1) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(0(4(x1))) 486.87/124.70 2^1(4(5(1(0(x1))))) -> 2^1(4(x1)) 486.87/124.70 The remaining pairs can at least be oriented weakly. 486.87/124.70 Used ordering: Matrix interpretation [MATRO] to (N^3, +, *, >=, >) : 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(2^1(x_1)) = [[0]] + [[1, 0, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(0(x_1)) = [[1], [0], [0]] + [[1, 0, 2], [0, 0, 0], [0, 0, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(1(x_1)) = [[0], [0], [0]] + [[3, 0, 0], [1, 0, 0], [2, 0, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(2(x_1)) = [[0], [0], [0]] + [[3, 0, 0], [0, 0, 0], [0, 0, 2]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(4(x_1)) = [[0], [0], [0]] + [[0, 0, 1], [0, 0, 0], [0, 0, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(3(x_1)) = [[0], [0], [0]] + [[0, 0, 0], [0, 0, 0], [0, 1, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 <<< 486.87/124.70 POL(5(x_1)) = [[0], [0], [0]] + [[0, 0, 0], [0, 0, 0], [0, 2, 0]] * x_1 486.87/124.70 >>> 486.87/124.70 486.87/124.70 486.87/124.70 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (10) 486.87/124.70 Obligation: 486.87/124.70 Q DP problem: 486.87/124.70 P is empty. 486.87/124.70 The TRS R consists of the following rules: 486.87/124.70 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(x1))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(1(3(0(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(1(3(0(2(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 0(3(0(1(3(2(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(3(0(1(4(x1)))))) 486.87/124.70 2(0(1(0(x1)))) -> 2(0(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(3(0(4(2(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(3(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(4(x1))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(4(1(3(0(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(4(1(3(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 3(0(1(3(4(2(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 1(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 4(0(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 0(4(1(3(2(3(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(4(1(3(0(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(4(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(1(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(5(0(3(1(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(3(4(x1)))))) 486.87/124.70 2(4(1(0(x1)))) -> 2(1(3(0(4(4(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 4(0(2(0(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 2(0(0(4(0(1(x1)))))) 486.87/124.70 2(0(1(0(0(x1))))) -> 0(0(3(0(1(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(0(1(3(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 1(3(0(4(2(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(2(0(1(3(0(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 2(0(4(3(0(1(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 4(1(3(0(0(2(x1)))))) 486.87/124.70 2(4(1(0(0(x1))))) -> 0(0(4(0(1(2(x1)))))) 486.87/124.70 2(0(2(1(0(x1))))) -> 2(4(0(1(0(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(0(1(3(2(x1)))))) 486.87/124.70 2(4(2(1(0(x1))))) -> 4(2(2(0(1(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(4(1(3(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 1(4(0(3(4(2(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 0(4(0(1(2(3(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(1(3(3(0(4(x1)))))) 486.87/124.70 2(4(3(1(0(x1))))) -> 2(3(0(4(1(4(x1)))))) 486.87/124.70 2(0(4(1(0(x1))))) -> 4(0(1(3(0(2(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 5(0(1(3(2(0(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(1(5(0(3(x1)))))) 486.87/124.70 2(0(5(1(0(x1))))) -> 2(0(0(3(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(4(0(1(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(5(4(4(0(2(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(1(3(2(0(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(5(2(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(5(0(1(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(3(1(2(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(2(0(1(3(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(1(5(0(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 5(0(1(2(4(4(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 1(2(5(4(0(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 3(4(0(2(1(5(x1)))))) 486.87/124.70 2(4(5(1(0(x1))))) -> 2(4(0(1(3(5(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(1(4(4(0(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 0(4(1(2(2(3(x1)))))) 486.87/124.70 2(4(1(2(0(x1))))) -> 2(2(3(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 1(5(3(4(0(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 5(1(3(0(4(2(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 2(3(5(0(1(4(x1)))))) 486.87/124.70 2(4(1(0(5(x1))))) -> 4(0(1(1(2(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(3(0(5(x1)))))) 486.87/124.70 2(0(2(0(5(x1))))) -> 2(2(0(4(0(5(x1)))))) 486.87/124.70 2(4(2(0(5(x1))))) -> 2(2(3(0(4(5(x1)))))) 486.87/124.70 2(0(5(1(5(x1))))) -> 0(5(4(1(2(5(x1)))))) 486.87/124.70 2(4(5(1(5(x1))))) -> 5(4(0(1(2(5(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 5(4(4(1(2(3(x1)))))) 486.87/124.70 2(4(1(4(5(x1))))) -> 2(1(5(3(4(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 4(2(2(5(0(3(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 2(2(3(5(0(4(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(2(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(3(5(x1)))))) 486.87/124.70 2(0(2(4(5(x1))))) -> 0(4(2(2(4(5(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(5(4(4(0(x1)))))) 486.87/124.70 2(4(2(4(5(x1))))) -> 2(2(3(4(4(5(x1)))))) 486.87/124.70 2(4(5(4(5(x1))))) -> 2(5(4(0(5(4(x1)))))) 486.87/124.70 486.87/124.70 Q is empty. 486.87/124.70 We have to consider all minimal (P,Q,R)-chains. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (11) PisEmptyProof (EQUIVALENT) 486.87/124.70 The TRS P is empty. Hence, there is no (P,Q,R) chain. 486.87/124.70 ---------------------------------------- 486.87/124.70 486.87/124.70 (12) 486.87/124.70 YES 487.08/124.80 EOF