72.89/19.33 YES 72.89/19.38 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 72.89/19.38 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 72.89/19.38 72.89/19.38 72.89/19.38 Termination w.r.t. Q of the given QTRS could be proven: 72.89/19.38 72.89/19.38 (0) QTRS 72.89/19.38 (1) QTRS Reverse [EQUIVALENT, 0 ms] 72.89/19.38 (2) QTRS 72.89/19.38 (3) DependencyPairsProof [EQUIVALENT, 262 ms] 72.89/19.38 (4) QDP 72.89/19.38 (5) DependencyGraphProof [EQUIVALENT, 7 ms] 72.89/19.38 (6) QDP 72.89/19.38 (7) QDPOrderProof [EQUIVALENT, 341 ms] 72.89/19.38 (8) QDP 72.89/19.38 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 72.89/19.38 (10) AND 72.89/19.38 (11) QDP 72.89/19.38 (12) QDPOrderProof [EQUIVALENT, 0 ms] 72.89/19.38 (13) QDP 72.89/19.38 (14) QDPOrderProof [EQUIVALENT, 53 ms] 72.89/19.38 (15) QDP 72.89/19.38 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 72.89/19.38 (17) TRUE 72.89/19.38 (18) QDP 72.89/19.38 (19) QDPOrderProof [EQUIVALENT, 0 ms] 72.89/19.38 (20) QDP 72.89/19.38 (21) QDPOrderProof [EQUIVALENT, 46 ms] 72.89/19.38 (22) QDP 72.89/19.38 (23) PisEmptyProof [EQUIVALENT, 0 ms] 72.89/19.38 (24) YES 72.89/19.38 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (0) 72.89/19.38 Obligation: 72.89/19.38 Q restricted rewrite system: 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 0(0(1(x1))) -> 2(3(1(4(0(0(x1)))))) 72.89/19.38 1(0(0(x1))) -> 0(2(3(1(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 0(2(1(2(3(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(0(5(2(3(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 2(1(0(2(3(1(x1)))))) 72.89/19.38 4(0(0(x1))) -> 0(2(3(4(0(x1))))) 72.89/19.38 4(0(1(x1))) -> 0(2(2(1(4(x1))))) 72.89/19.38 4(0(1(x1))) -> 0(2(3(4(1(x1))))) 72.89/19.38 5(1(0(x1))) -> 5(2(3(1(0(x1))))) 72.89/19.38 5(1(0(x1))) -> 5(2(3(3(1(0(x1)))))) 72.89/19.38 5(1(1(x1))) -> 2(3(1(4(1(5(x1)))))) 72.89/19.38 5(1(1(x1))) -> 5(3(2(3(1(1(x1)))))) 72.89/19.38 0(0(3(0(x1)))) -> 4(0(0(2(3(0(x1)))))) 72.89/19.38 0(3(0(1(x1)))) -> 0(0(2(2(1(3(x1)))))) 72.89/19.38 1(0(3(1(x1)))) -> 0(1(2(3(1(3(x1)))))) 72.89/19.38 1(0(3(1(x1)))) -> 0(4(1(2(3(1(x1)))))) 72.89/19.38 1(2(5(1(x1)))) -> 5(4(2(3(1(1(x1)))))) 72.89/19.38 4(0(1(1(x1)))) -> 0(2(4(1(1(x1))))) 72.89/19.38 4(0(1(5(x1)))) -> 5(0(2(3(1(4(x1)))))) 72.89/19.38 4(0(1(5(x1)))) -> 5(4(1(3(4(0(x1)))))) 72.89/19.38 4(0(3(0(x1)))) -> 0(2(3(0(4(5(x1)))))) 72.89/19.38 4(0(3(5(x1)))) -> 0(2(3(4(1(5(x1)))))) 72.89/19.38 4(2(5(0(x1)))) -> 4(5(2(3(0(x1))))) 72.89/19.38 4(2(5(0(x1)))) -> 4(0(2(3(5(4(x1)))))) 72.89/19.38 4(3(0(0(x1)))) -> 0(2(3(4(3(0(x1)))))) 72.89/19.38 4(3(0(1(x1)))) -> 1(4(0(2(3(x1))))) 72.89/19.38 4(3(0(1(x1)))) -> 3(0(2(2(4(1(x1)))))) 72.89/19.38 5(1(1(1(x1)))) -> 2(2(1(1(5(1(x1)))))) 72.89/19.38 0(0(1(2(5(x1))))) -> 2(3(1(5(0(0(x1)))))) 72.89/19.38 0(1(3(0(0(x1))))) -> 2(3(1(0(0(0(x1)))))) 72.89/19.38 0(2(5(3(1(x1))))) -> 2(0(5(4(1(3(x1)))))) 72.89/19.38 0(2(5(5(1(x1))))) -> 0(2(1(5(4(5(x1)))))) 72.89/19.38 0(3(0(5(1(x1))))) -> 2(1(5(3(0(0(x1)))))) 72.89/19.38 0(4(3(5(1(x1))))) -> 2(4(3(1(0(5(x1)))))) 72.89/19.38 0(5(1(1(5(x1))))) -> 0(1(2(1(5(5(x1)))))) 72.89/19.38 0(5(1(3(0(x1))))) -> 2(1(3(5(0(0(x1)))))) 72.89/19.38 1(0(4(1(5(x1))))) -> 4(1(2(1(5(0(x1)))))) 72.89/19.38 1(2(4(5(1(x1))))) -> 1(4(1(5(2(3(x1)))))) 72.89/19.38 1(2(5(3(0(x1))))) -> 1(4(2(3(5(0(x1)))))) 72.89/19.38 1(3(0(4(0(x1))))) -> 0(2(3(1(4(0(x1)))))) 72.89/19.38 4(0(3(1(5(x1))))) -> 4(4(3(5(0(1(x1)))))) 72.89/19.38 4(2(5(0(0(x1))))) -> 0(5(2(0(4(4(x1)))))) 72.89/19.38 4(2(5(0(0(x1))))) -> 5(2(3(4(0(0(x1)))))) 72.89/19.38 4(2(5(3(0(x1))))) -> 4(0(2(5(2(3(x1)))))) 72.89/19.38 4(2(5(3(0(x1))))) -> 5(2(4(3(1(0(x1)))))) 72.89/19.38 4(2(5(3(1(x1))))) -> 2(3(1(5(4(2(x1)))))) 72.89/19.38 4(3(0(0(1(x1))))) -> 2(4(3(1(0(0(x1)))))) 72.89/19.38 4(3(2(5(0(x1))))) -> 0(4(2(3(4(5(x1)))))) 72.89/19.38 4(5(1(3(1(x1))))) -> 4(1(3(5(4(1(x1)))))) 72.89/19.38 5(1(4(3(1(x1))))) -> 5(3(2(4(1(1(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (1) QTRS Reverse (EQUIVALENT) 72.89/19.38 We applied the QTRS Reverse Processor [REVERSE]. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (2) 72.89/19.38 Obligation: 72.89/19.38 Q restricted rewrite system: 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (3) DependencyPairsProof (EQUIVALENT) 72.89/19.38 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (4) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 1^1(0(0(x1))) -> 0^1(0(4(1(3(2(x1)))))) 72.89/19.38 1^1(0(0(x1))) -> 0^1(4(1(3(2(x1))))) 72.89/19.38 1^1(0(0(x1))) -> 1^1(3(2(x1))) 72.89/19.38 0^1(0(1(x1))) -> 0^1(1(3(2(0(x1))))) 72.89/19.38 0^1(0(1(x1))) -> 1^1(3(2(0(x1)))) 72.89/19.38 0^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(1(x1))) -> 1^1(3(2(1(2(0(x1)))))) 72.89/19.38 1^1(0(1(x1))) -> 1^1(2(0(x1))) 72.89/19.38 1^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(1(x1))) -> 1^1(3(2(5(0(1(x1)))))) 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 1^1(0(1(x1))) -> 1^1(3(2(0(1(2(x1)))))) 72.89/19.38 1^1(0(1(x1))) -> 0^1(1(2(x1))) 72.89/19.38 1^1(0(1(x1))) -> 1^1(2(x1)) 72.89/19.38 0^1(0(4(x1))) -> 0^1(4(3(2(0(x1))))) 72.89/19.38 0^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(4(x1))) -> 1^1(2(2(0(x1)))) 72.89/19.38 1^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(4(x1))) -> 1^1(4(3(2(0(x1))))) 72.89/19.38 0^1(1(5(x1))) -> 0^1(1(3(2(5(x1))))) 72.89/19.38 0^1(1(5(x1))) -> 1^1(3(2(5(x1)))) 72.89/19.38 0^1(1(5(x1))) -> 0^1(1(3(3(2(5(x1)))))) 72.89/19.38 0^1(1(5(x1))) -> 1^1(3(3(2(5(x1))))) 72.89/19.38 1^1(1(5(x1))) -> 5^1(1(4(1(3(2(x1)))))) 72.89/19.38 1^1(1(5(x1))) -> 1^1(4(1(3(2(x1))))) 72.89/19.38 1^1(1(5(x1))) -> 1^1(3(2(x1))) 72.89/19.38 1^1(1(5(x1))) -> 1^1(1(3(2(3(5(x1)))))) 72.89/19.38 1^1(1(5(x1))) -> 1^1(3(2(3(5(x1))))) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(3(2(0(0(4(x1)))))) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(0(4(x1))) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(4(x1)) 72.89/19.38 1^1(0(3(0(x1)))) -> 1^1(2(2(0(0(x1))))) 72.89/19.38 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(3(2(1(0(x1))))) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 0^1(x1) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(3(2(1(4(0(x1)))))) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(4(0(x1))) 72.89/19.38 1^1(5(2(1(x1)))) -> 1^1(1(3(2(4(5(x1)))))) 72.89/19.38 1^1(5(2(1(x1)))) -> 1^1(3(2(4(5(x1))))) 72.89/19.38 1^1(5(2(1(x1)))) -> 5^1(x1) 72.89/19.38 1^1(1(0(4(x1)))) -> 1^1(1(4(2(0(x1))))) 72.89/19.38 1^1(1(0(4(x1)))) -> 1^1(4(2(0(x1)))) 72.89/19.38 1^1(1(0(4(x1)))) -> 0^1(x1) 72.89/19.38 5^1(1(0(4(x1)))) -> 1^1(3(2(0(5(x1))))) 72.89/19.38 5^1(1(0(4(x1)))) -> 0^1(5(x1)) 72.89/19.38 5^1(1(0(4(x1)))) -> 5^1(x1) 72.89/19.38 5^1(1(0(4(x1)))) -> 0^1(4(3(1(4(5(x1)))))) 72.89/19.38 5^1(1(0(4(x1)))) -> 1^1(4(5(x1))) 72.89/19.38 0^1(3(0(4(x1)))) -> 5^1(4(0(3(2(0(x1)))))) 72.89/19.38 0^1(3(0(4(x1)))) -> 0^1(3(2(0(x1)))) 72.89/19.38 0^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 5^1(3(0(4(x1)))) -> 5^1(1(4(3(2(0(x1)))))) 72.89/19.38 5^1(3(0(4(x1)))) -> 1^1(4(3(2(0(x1))))) 72.89/19.38 5^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(5(2(4(x1)))) -> 0^1(3(2(5(4(x1))))) 72.89/19.38 0^1(5(2(4(x1)))) -> 5^1(4(x1)) 72.89/19.38 0^1(5(2(4(x1)))) -> 5^1(3(2(0(4(x1))))) 72.89/19.38 0^1(5(2(4(x1)))) -> 0^1(4(x1)) 72.89/19.38 0^1(0(3(4(x1)))) -> 0^1(3(4(3(2(0(x1)))))) 72.89/19.38 0^1(0(3(4(x1)))) -> 0^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(4(1(x1))) 72.89/19.38 1^1(0(3(4(x1)))) -> 1^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 1^1(4(2(2(0(3(x1)))))) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(3(x1)) 72.89/19.38 1^1(1(1(5(x1)))) -> 1^1(5(1(1(2(2(x1)))))) 72.89/19.38 1^1(1(1(5(x1)))) -> 5^1(1(1(2(2(x1))))) 72.89/19.38 1^1(1(1(5(x1)))) -> 1^1(1(2(2(x1)))) 72.89/19.38 1^1(1(1(5(x1)))) -> 1^1(2(2(x1))) 72.89/19.38 5^1(2(1(0(0(x1))))) -> 0^1(0(5(1(3(2(x1)))))) 72.89/19.38 5^1(2(1(0(0(x1))))) -> 0^1(5(1(3(2(x1))))) 72.89/19.38 5^1(2(1(0(0(x1))))) -> 5^1(1(3(2(x1)))) 72.89/19.38 5^1(2(1(0(0(x1))))) -> 1^1(3(2(x1))) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(1(3(2(x1))))) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(1(3(2(x1)))) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 1^1(3(2(x1))) 72.89/19.38 1^1(3(5(2(0(x1))))) -> 1^1(4(5(0(2(x1))))) 72.89/19.38 1^1(3(5(2(0(x1))))) -> 5^1(0(2(x1))) 72.89/19.38 1^1(3(5(2(0(x1))))) -> 0^1(2(x1)) 72.89/19.38 1^1(5(5(2(0(x1))))) -> 5^1(4(5(1(2(0(x1)))))) 72.89/19.38 1^1(5(5(2(0(x1))))) -> 5^1(1(2(0(x1)))) 72.89/19.38 1^1(5(5(2(0(x1))))) -> 1^1(2(0(x1))) 72.89/19.38 1^1(5(0(3(0(x1))))) -> 0^1(0(3(5(1(2(x1)))))) 72.89/19.38 1^1(5(0(3(0(x1))))) -> 0^1(3(5(1(2(x1))))) 72.89/19.38 1^1(5(0(3(0(x1))))) -> 5^1(1(2(x1))) 72.89/19.38 1^1(5(0(3(0(x1))))) -> 1^1(2(x1)) 72.89/19.38 1^1(5(3(4(0(x1))))) -> 5^1(0(1(3(4(2(x1)))))) 72.89/19.38 1^1(5(3(4(0(x1))))) -> 0^1(1(3(4(2(x1))))) 72.89/19.38 1^1(5(3(4(0(x1))))) -> 1^1(3(4(2(x1)))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 5^1(5(1(2(1(0(x1)))))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 5^1(1(2(1(0(x1))))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(2(1(0(x1)))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 0^1(3(1(5(0(x1))))) -> 0^1(0(5(3(1(2(x1)))))) 72.89/19.38 0^1(3(1(5(0(x1))))) -> 0^1(5(3(1(2(x1))))) 72.89/19.38 0^1(3(1(5(0(x1))))) -> 5^1(3(1(2(x1)))) 72.89/19.38 0^1(3(1(5(0(x1))))) -> 1^1(2(x1)) 72.89/19.38 5^1(1(4(0(1(x1))))) -> 0^1(5(1(2(1(4(x1)))))) 72.89/19.38 5^1(1(4(0(1(x1))))) -> 5^1(1(2(1(4(x1))))) 72.89/19.38 5^1(1(4(0(1(x1))))) -> 1^1(2(1(4(x1)))) 72.89/19.38 5^1(1(4(0(1(x1))))) -> 1^1(4(x1)) 72.89/19.38 1^1(5(4(2(1(x1))))) -> 5^1(1(4(1(x1)))) 72.89/19.38 1^1(5(4(2(1(x1))))) -> 1^1(4(1(x1))) 72.89/19.38 0^1(3(5(2(1(x1))))) -> 0^1(5(3(2(4(1(x1)))))) 72.89/19.38 0^1(3(5(2(1(x1))))) -> 5^1(3(2(4(1(x1))))) 72.89/19.38 0^1(4(0(3(1(x1))))) -> 0^1(4(1(3(2(0(x1)))))) 72.89/19.38 0^1(4(0(3(1(x1))))) -> 1^1(3(2(0(x1)))) 72.89/19.38 0^1(4(0(3(1(x1))))) -> 0^1(x1) 72.89/19.38 5^1(1(3(0(4(x1))))) -> 1^1(0(5(3(4(4(x1)))))) 72.89/19.38 5^1(1(3(0(4(x1))))) -> 0^1(5(3(4(4(x1))))) 72.89/19.38 5^1(1(3(0(4(x1))))) -> 5^1(3(4(4(x1)))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(2(5(0(x1)))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(0(x1)) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(x1) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(0(4(3(2(5(x1)))))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(4(3(2(5(x1))))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 5^1(2(0(4(x1)))) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 0^1(4(x1)) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 0^1(1(3(4(2(5(x1)))))) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 1^1(3(4(2(5(x1))))) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 1^1(3(5(2(4(x1))))) -> 5^1(1(3(2(x1)))) 72.89/19.38 1^1(3(5(2(4(x1))))) -> 1^1(3(2(x1))) 72.89/19.38 1^1(0(0(3(4(x1))))) -> 0^1(0(1(3(4(2(x1)))))) 72.89/19.38 1^1(0(0(3(4(x1))))) -> 0^1(1(3(4(2(x1))))) 72.89/19.38 1^1(0(0(3(4(x1))))) -> 1^1(3(4(2(x1)))) 72.89/19.38 0^1(5(2(3(4(x1))))) -> 5^1(4(3(2(4(0(x1)))))) 72.89/19.38 0^1(5(2(3(4(x1))))) -> 0^1(x1) 72.89/19.38 1^1(3(1(5(4(x1))))) -> 1^1(4(5(3(1(4(x1)))))) 72.89/19.38 1^1(3(1(5(4(x1))))) -> 5^1(3(1(4(x1)))) 72.89/19.38 1^1(3(1(5(4(x1))))) -> 1^1(4(x1)) 72.89/19.38 1^1(3(4(1(5(x1))))) -> 1^1(1(4(2(3(5(x1)))))) 72.89/19.38 1^1(3(4(1(5(x1))))) -> 1^1(4(2(3(5(x1))))) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 We have to consider all minimal (P,Q,R)-chains. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (5) DependencyGraphProof (EQUIVALENT) 72.89/19.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 101 less nodes. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (6) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 0^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 0^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(0(4(x1))) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(4(x1)) 72.89/19.38 0^1(4(0(3(1(x1))))) -> 0^1(x1) 72.89/19.38 0^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(5(2(4(x1)))) -> 0^1(4(x1)) 72.89/19.38 0^1(0(3(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(1(3(2(x1))))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(0(x1)) 72.89/19.38 5^1(1(0(4(x1)))) -> 0^1(5(x1)) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(x1) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(0(4(3(2(5(x1)))))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 5^1(1(0(4(x1)))) -> 5^1(x1) 72.89/19.38 5^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 0^1(4(x1)) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 1^1(0(0(x1))) -> 0^1(0(4(1(3(2(x1)))))) 72.89/19.38 1^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 0^1(5(2(3(4(x1))))) -> 0^1(x1) 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 1^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 0^1(x1) 72.89/19.38 1^1(5(2(1(x1)))) -> 5^1(x1) 72.89/19.38 1^1(1(0(4(x1)))) -> 0^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(4(1(x1))) 72.89/19.38 1^1(0(3(4(x1)))) -> 1^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(3(x1)) 72.89/19.38 1^1(0(0(3(4(x1))))) -> 0^1(0(1(3(4(2(x1)))))) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 We have to consider all minimal (P,Q,R)-chains. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (7) QDPOrderProof (EQUIVALENT) 72.89/19.38 We use the reduction pair processor [LPAR04,JAR06]. 72.89/19.38 72.89/19.38 72.89/19.38 The following pairs can be oriented strictly and are deleted. 72.89/19.38 72.89/19.38 0^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 0^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(0(4(x1))) 72.89/19.38 0^1(3(0(0(x1)))) -> 0^1(4(x1)) 72.89/19.38 0^1(4(0(3(1(x1))))) -> 0^1(x1) 72.89/19.38 0^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(0(3(4(x1)))) -> 0^1(x1) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(1(3(2(x1))))) 72.89/19.38 5^1(1(0(4(x1)))) -> 0^1(5(x1)) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(x1) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 5^1(1(0(4(x1)))) -> 5^1(x1) 72.89/19.38 5^1(3(0(4(x1)))) -> 0^1(x1) 72.89/19.38 1^1(0(0(x1))) -> 0^1(0(4(1(3(2(x1)))))) 72.89/19.38 1^1(0(1(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(4(x1))) -> 0^1(x1) 72.89/19.38 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 0^1(x1) 72.89/19.38 1^1(1(0(4(x1)))) -> 0^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(4(1(x1))) 72.89/19.38 1^1(0(3(4(x1)))) -> 1^1(x1) 72.89/19.38 1^1(0(3(4(x1)))) -> 0^1(3(x1)) 72.89/19.38 1^1(0(0(3(4(x1))))) -> 0^1(0(1(3(4(2(x1)))))) 72.89/19.38 The remaining pairs can at least be oriented weakly. 72.89/19.38 Used ordering: Polynomial interpretation [POLO]: 72.89/19.38 72.89/19.38 POL(0(x_1)) = 1 + x_1 72.89/19.38 POL(0^1(x_1)) = x_1 72.89/19.38 POL(1(x_1)) = x_1 72.89/19.38 POL(1^1(x_1)) = x_1 72.89/19.38 POL(2(x_1)) = x_1 72.89/19.38 POL(3(x_1)) = x_1 72.89/19.38 POL(4(x_1)) = x_1 72.89/19.38 POL(5(x_1)) = x_1 72.89/19.38 POL(5^1(x_1)) = x_1 72.89/19.38 72.89/19.38 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 72.89/19.38 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (8) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 0^1(5(2(4(x1)))) -> 0^1(4(x1)) 72.89/19.38 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 5^1(0(x1)) 72.89/19.38 0^1(0(5(2(4(x1))))) -> 0^1(0(4(3(2(5(x1)))))) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 0^1(4(x1)) 72.89/19.38 0^1(3(5(2(4(x1))))) -> 5^1(x1) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 0^1(5(2(3(4(x1))))) -> 0^1(x1) 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 1^1(5(2(1(x1)))) -> 5^1(x1) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 We have to consider all minimal (P,Q,R)-chains. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (9) DependencyGraphProof (EQUIVALENT) 72.89/19.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (10) 72.89/19.38 Complex Obligation (AND) 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (11) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 1^1(5(2(1(x1)))) -> 5^1(x1) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 We have to consider all minimal (P,Q,R)-chains. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (12) QDPOrderProof (EQUIVALENT) 72.89/19.38 We use the reduction pair processor [LPAR04,JAR06]. 72.89/19.38 72.89/19.38 72.89/19.38 The following pairs can be oriented strictly and are deleted. 72.89/19.38 72.89/19.38 1^1(5(2(1(x1)))) -> 5^1(x1) 72.89/19.38 The remaining pairs can at least be oriented weakly. 72.89/19.38 Used ordering: Polynomial interpretation [POLO]: 72.89/19.38 72.89/19.38 POL(0(x_1)) = 0 72.89/19.38 POL(1(x_1)) = 0 72.89/19.38 POL(1^1(x_1)) = 1 + x_1 72.89/19.38 POL(2(x_1)) = 1 72.89/19.38 POL(3(x_1)) = 0 72.89/19.38 POL(4(x_1)) = 0 72.89/19.38 POL(5(x_1)) = x_1 72.89/19.38 POL(5^1(x_1)) = 1 72.89/19.38 72.89/19.38 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 72.89/19.38 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (13) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.38 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.38 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.38 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.38 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.38 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.38 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.38 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.38 72.89/19.38 Q is empty. 72.89/19.38 We have to consider all minimal (P,Q,R)-chains. 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (14) QDPOrderProof (EQUIVALENT) 72.89/19.38 We use the reduction pair processor [LPAR04,JAR06]. 72.89/19.38 72.89/19.38 72.89/19.38 The following pairs can be oriented strictly and are deleted. 72.89/19.38 72.89/19.38 5^1(1(1(5(0(x1))))) -> 1^1(0(x1)) 72.89/19.38 1^1(3(0(1(x1)))) -> 1^1(0(x1)) 72.89/19.38 The remaining pairs can at least be oriented weakly. 72.89/19.38 Used ordering: Polynomial interpretation [POLO]: 72.89/19.38 72.89/19.38 POL(0(x_1)) = 1 72.89/19.38 POL(1(x_1)) = 1 + x_1 72.89/19.38 POL(1^1(x_1)) = x_1 72.89/19.38 POL(2(x_1)) = 0 72.89/19.38 POL(3(x_1)) = 1 + x_1 72.89/19.38 POL(4(x_1)) = 0 72.89/19.38 POL(5(x_1)) = 0 72.89/19.38 POL(5^1(x_1)) = x_1 72.89/19.38 72.89/19.38 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 72.89/19.38 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.38 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.38 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.38 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.38 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.38 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.38 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.38 72.89/19.38 72.89/19.38 ---------------------------------------- 72.89/19.38 72.89/19.38 (15) 72.89/19.38 Obligation: 72.89/19.38 Q DP problem: 72.89/19.38 The TRS P consists of the following rules: 72.89/19.38 72.89/19.38 1^1(0(1(x1))) -> 5^1(0(1(x1))) 72.89/19.38 72.89/19.38 The TRS R consists of the following rules: 72.89/19.38 72.89/19.38 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.38 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.38 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.38 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.38 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.38 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.38 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.38 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.38 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.38 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.38 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.38 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.38 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.38 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.38 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.38 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.38 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.38 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.38 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.38 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.38 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.38 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.38 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.38 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.38 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.39 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.39 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.39 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.39 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.39 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.39 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.39 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.39 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.39 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.39 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.39 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.39 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.39 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.39 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.39 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.39 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.39 72.89/19.39 Q is empty. 72.89/19.39 We have to consider all minimal (P,Q,R)-chains. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (16) DependencyGraphProof (EQUIVALENT) 72.89/19.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (17) 72.89/19.39 TRUE 72.89/19.39 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (18) 72.89/19.39 Obligation: 72.89/19.39 Q DP problem: 72.89/19.39 The TRS P consists of the following rules: 72.89/19.39 72.89/19.39 0^1(5(2(3(4(x1))))) -> 0^1(x1) 72.89/19.39 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.39 72.89/19.39 The TRS R consists of the following rules: 72.89/19.39 72.89/19.39 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.39 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.39 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.39 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.39 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.39 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.39 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.39 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.39 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.39 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.39 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.39 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.39 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.39 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.39 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.39 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.39 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.39 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.39 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.39 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.39 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.39 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.39 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.39 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.39 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.39 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.39 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.39 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.39 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.39 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.39 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.39 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.39 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.39 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.39 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.39 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.39 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.39 72.89/19.39 Q is empty. 72.89/19.39 We have to consider all minimal (P,Q,R)-chains. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (19) QDPOrderProof (EQUIVALENT) 72.89/19.39 We use the reduction pair processor [LPAR04,JAR06]. 72.89/19.39 72.89/19.39 72.89/19.39 The following pairs can be oriented strictly and are deleted. 72.89/19.39 72.89/19.39 0^1(5(2(3(4(x1))))) -> 0^1(x1) 72.89/19.39 The remaining pairs can at least be oriented weakly. 72.89/19.39 Used ordering: Polynomial interpretation [POLO]: 72.89/19.39 72.89/19.39 POL(0(x_1)) = 0 72.89/19.39 POL(0^1(x_1)) = x_1 72.89/19.39 POL(1(x_1)) = 0 72.89/19.39 POL(2(x_1)) = 1 + x_1 72.89/19.39 POL(3(x_1)) = x_1 72.89/19.39 POL(4(x_1)) = x_1 72.89/19.39 POL(5(x_1)) = x_1 72.89/19.39 72.89/19.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 72.89/19.39 72.89/19.39 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.39 72.89/19.39 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (20) 72.89/19.39 Obligation: 72.89/19.39 Q DP problem: 72.89/19.39 The TRS P consists of the following rules: 72.89/19.39 72.89/19.39 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.39 72.89/19.39 The TRS R consists of the following rules: 72.89/19.39 72.89/19.39 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.39 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.39 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.39 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.39 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.39 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.39 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.39 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.39 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.39 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.39 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.39 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.39 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.39 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.39 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.39 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.39 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.39 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.39 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.39 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.39 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.39 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.39 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.39 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.39 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.39 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.39 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.39 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.39 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.39 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.39 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.39 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.39 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.39 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.39 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.39 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.39 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.39 72.89/19.39 Q is empty. 72.89/19.39 We have to consider all minimal (P,Q,R)-chains. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (21) QDPOrderProof (EQUIVALENT) 72.89/19.39 We use the reduction pair processor [LPAR04,JAR06]. 72.89/19.39 72.89/19.39 72.89/19.39 The following pairs can be oriented strictly and are deleted. 72.89/19.39 72.89/19.39 0^1(0(3(1(0(x1))))) -> 0^1(0(0(1(3(2(x1)))))) 72.89/19.39 The remaining pairs can at least be oriented weakly. 72.89/19.39 Used ordering: Polynomial interpretation [POLO]: 72.89/19.39 72.89/19.39 POL(0(x_1)) = x_1 72.89/19.39 POL(0^1(x_1)) = x_1 72.89/19.39 POL(1(x_1)) = 0 72.89/19.39 POL(2(x_1)) = 0 72.89/19.39 POL(3(x_1)) = 1 72.89/19.39 POL(4(x_1)) = 0 72.89/19.39 POL(5(x_1)) = 1 72.89/19.39 72.89/19.39 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 72.89/19.39 72.89/19.39 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.39 72.89/19.39 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (22) 72.89/19.39 Obligation: 72.89/19.39 Q DP problem: 72.89/19.39 P is empty. 72.89/19.39 The TRS R consists of the following rules: 72.89/19.39 72.89/19.39 1(0(0(x1))) -> 0(0(4(1(3(2(x1)))))) 72.89/19.39 0(0(1(x1))) -> 0(1(3(2(0(x1))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(1(2(0(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(5(0(1(x1)))))) 72.89/19.39 1(0(1(x1))) -> 1(3(2(0(1(2(x1)))))) 72.89/19.39 0(0(4(x1))) -> 0(4(3(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 4(1(2(2(0(x1))))) 72.89/19.39 1(0(4(x1))) -> 1(4(3(2(0(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(2(5(x1))))) 72.89/19.39 0(1(5(x1))) -> 0(1(3(3(2(5(x1)))))) 72.89/19.39 1(1(5(x1))) -> 5(1(4(1(3(2(x1)))))) 72.89/19.39 1(1(5(x1))) -> 1(1(3(2(3(5(x1)))))) 72.89/19.39 0(3(0(0(x1)))) -> 0(3(2(0(0(4(x1)))))) 72.89/19.39 1(0(3(0(x1)))) -> 3(1(2(2(0(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 3(1(3(2(1(0(x1)))))) 72.89/19.39 1(3(0(1(x1)))) -> 1(3(2(1(4(0(x1)))))) 72.89/19.39 1(5(2(1(x1)))) -> 1(1(3(2(4(5(x1)))))) 72.89/19.39 1(1(0(4(x1)))) -> 1(1(4(2(0(x1))))) 72.89/19.39 5(1(0(4(x1)))) -> 4(1(3(2(0(5(x1)))))) 72.89/19.39 5(1(0(4(x1)))) -> 0(4(3(1(4(5(x1)))))) 72.89/19.39 0(3(0(4(x1)))) -> 5(4(0(3(2(0(x1)))))) 72.89/19.39 5(3(0(4(x1)))) -> 5(1(4(3(2(0(x1)))))) 72.89/19.39 0(5(2(4(x1)))) -> 0(3(2(5(4(x1))))) 72.89/19.39 0(5(2(4(x1)))) -> 4(5(3(2(0(4(x1)))))) 72.89/19.39 0(0(3(4(x1)))) -> 0(3(4(3(2(0(x1)))))) 72.89/19.39 1(0(3(4(x1)))) -> 3(2(0(4(1(x1))))) 72.89/19.39 1(0(3(4(x1)))) -> 1(4(2(2(0(3(x1)))))) 72.89/19.39 1(1(1(5(x1)))) -> 1(5(1(1(2(2(x1)))))) 72.89/19.39 5(2(1(0(0(x1))))) -> 0(0(5(1(3(2(x1)))))) 72.89/19.39 0(0(3(1(0(x1))))) -> 0(0(0(1(3(2(x1)))))) 72.89/19.39 1(3(5(2(0(x1))))) -> 3(1(4(5(0(2(x1)))))) 72.89/19.39 1(5(5(2(0(x1))))) -> 5(4(5(1(2(0(x1)))))) 72.89/19.39 1(5(0(3(0(x1))))) -> 0(0(3(5(1(2(x1)))))) 72.89/19.39 1(5(3(4(0(x1))))) -> 5(0(1(3(4(2(x1)))))) 72.89/19.39 5(1(1(5(0(x1))))) -> 5(5(1(2(1(0(x1)))))) 72.89/19.39 0(3(1(5(0(x1))))) -> 0(0(5(3(1(2(x1)))))) 72.89/19.39 5(1(4(0(1(x1))))) -> 0(5(1(2(1(4(x1)))))) 72.89/19.39 1(5(4(2(1(x1))))) -> 3(2(5(1(4(1(x1)))))) 72.89/19.39 0(3(5(2(1(x1))))) -> 0(5(3(2(4(1(x1)))))) 72.89/19.39 0(4(0(3(1(x1))))) -> 0(4(1(3(2(0(x1)))))) 72.89/19.39 5(1(3(0(4(x1))))) -> 1(0(5(3(4(4(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 4(4(0(2(5(0(x1)))))) 72.89/19.39 0(0(5(2(4(x1))))) -> 0(0(4(3(2(5(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 3(2(5(2(0(4(x1)))))) 72.89/19.39 0(3(5(2(4(x1))))) -> 0(1(3(4(2(5(x1)))))) 72.89/19.39 1(3(5(2(4(x1))))) -> 2(4(5(1(3(2(x1)))))) 72.89/19.39 1(0(0(3(4(x1))))) -> 0(0(1(3(4(2(x1)))))) 72.89/19.39 0(5(2(3(4(x1))))) -> 5(4(3(2(4(0(x1)))))) 72.89/19.39 1(3(1(5(4(x1))))) -> 1(4(5(3(1(4(x1)))))) 72.89/19.39 1(3(4(1(5(x1))))) -> 1(1(4(2(3(5(x1)))))) 72.89/19.39 72.89/19.39 Q is empty. 72.89/19.39 We have to consider all minimal (P,Q,R)-chains. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (23) PisEmptyProof (EQUIVALENT) 72.89/19.39 The TRS P is empty. Hence, there is no (P,Q,R) chain. 72.89/19.39 ---------------------------------------- 72.89/19.39 72.89/19.39 (24) 72.89/19.39 YES 73.23/19.44 EOF