56.92/15.33 YES 57.03/15.40 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 57.03/15.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 57.03/15.40 57.03/15.40 57.03/15.40 Termination w.r.t. Q of the given QTRS could be proven: 57.03/15.40 57.03/15.40 (0) QTRS 57.03/15.40 (1) QTRS Reverse [EQUIVALENT, 0 ms] 57.03/15.40 (2) QTRS 57.03/15.40 (3) DependencyPairsProof [EQUIVALENT, 300 ms] 57.03/15.40 (4) QDP 57.03/15.40 (5) DependencyGraphProof [EQUIVALENT, 8 ms] 57.03/15.40 (6) QDP 57.03/15.40 (7) QDPOrderProof [EQUIVALENT, 270 ms] 57.03/15.40 (8) QDP 57.03/15.40 (9) DependencyGraphProof [EQUIVALENT, 0 ms] 57.03/15.40 (10) AND 57.03/15.40 (11) QDP 57.03/15.40 (12) QDPOrderProof [EQUIVALENT, 61 ms] 57.03/15.40 (13) QDP 57.03/15.40 (14) DependencyGraphProof [EQUIVALENT, 0 ms] 57.03/15.40 (15) QDP 57.03/15.40 (16) QDPOrderProof [EQUIVALENT, 81 ms] 57.03/15.40 (17) QDP 57.03/15.40 (18) PisEmptyProof [EQUIVALENT, 0 ms] 57.03/15.40 (19) YES 57.03/15.40 (20) QDP 57.03/15.40 (21) QDPOrderProof [EQUIVALENT, 82 ms] 57.03/15.40 (22) QDP 57.03/15.40 (23) PisEmptyProof [EQUIVALENT, 0 ms] 57.03/15.40 (24) YES 57.03/15.40 57.03/15.40 57.03/15.40 ---------------------------------------- 57.03/15.40 57.03/15.40 (0) 57.03/15.40 Obligation: 57.03/15.40 Q restricted rewrite system: 57.03/15.40 The TRS R consists of the following rules: 57.03/15.40 57.03/15.40 0(0(1(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.40 0(0(1(x1))) -> 0(0(3(1(4(x1))))) 57.03/15.40 0(1(0(x1))) -> 0(0(1(2(4(x1))))) 57.03/15.40 0(1(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.40 0(1(0(x1))) -> 0(0(1(2(2(4(x1)))))) 57.03/15.40 0(1(0(x1))) -> 0(0(2(4(1(4(x1)))))) 57.03/15.40 0(4(0(x1))) -> 3(4(3(2(0(0(x1)))))) 57.03/15.40 0(0(1(0(x1)))) -> 0(0(0(1(4(x1))))) 57.03/15.40 0(0(4(1(x1)))) -> 0(0(2(1(4(x1))))) 57.03/15.40 0(0(4(1(x1)))) -> 0(0(2(1(4(3(x1)))))) 57.03/15.40 0(1(0(4(x1)))) -> 0(0(3(4(1(2(x1)))))) 57.03/15.40 0(1(3(0(x1)))) -> 0(0(3(1(4(x1))))) 57.03/15.40 0(1(3(0(x1)))) -> 0(3(0(1(2(x1))))) 57.03/15.40 0(1(3(0(x1)))) -> 0(0(1(3(2(5(x1)))))) 57.03/15.40 0(1(4(0(x1)))) -> 0(2(0(3(1(4(x1)))))) 57.03/15.40 0(1(5(1(x1)))) -> 2(1(1(4(5(0(x1)))))) 57.03/15.40 0(1(5(4(x1)))) -> 1(2(4(2(5(0(x1)))))) 57.03/15.40 0(1(5(4(x1)))) -> 4(1(0(3(2(5(x1)))))) 57.03/15.40 0(1(5(4(x1)))) -> 5(0(2(4(1(4(x1)))))) 57.03/15.40 0(3(0(1(x1)))) -> 0(0(3(1(2(2(x1)))))) 57.03/15.40 0(3(1(0(x1)))) -> 0(0(3(1(4(x1))))) 57.03/15.40 0(4(0(1(x1)))) -> 0(0(2(1(4(x1))))) 57.03/15.40 0(4(5(1(x1)))) -> 1(2(4(2(5(0(x1)))))) 57.03/15.40 0(4(5(1(x1)))) -> 3(1(4(5(0(2(x1)))))) 57.03/15.40 0(4(5(1(x1)))) -> 3(2(5(1(4(0(x1)))))) 57.03/15.40 0(4(5(1(x1)))) -> 5(3(0(5(1(4(x1)))))) 57.03/15.40 0(4(5(1(x1)))) -> 5(5(0(5(1(4(x1)))))) 57.03/15.40 0(4(5(4(x1)))) -> 5(2(4(4(0(4(x1)))))) 57.03/15.40 0(5(1(0(x1)))) -> 0(0(5(1(2(x1))))) 57.03/15.40 3(5(0(1(x1)))) -> 3(0(2(1(2(5(x1)))))) 57.03/15.40 3(5(1(0(x1)))) -> 0(5(1(3(2(x1))))) 57.03/15.40 3(5(1(0(x1)))) -> 2(1(2(0(5(3(x1)))))) 57.03/15.40 3(5(1(0(x1)))) -> 3(1(2(2(0(5(x1)))))) 57.03/15.40 3(5(1(0(x1)))) -> 5(1(3(2(0(2(x1)))))) 57.03/15.40 0(1(3(3(0(x1))))) -> 3(3(2(0(0(1(x1)))))) 57.03/15.40 0(1(3(5(1(x1))))) -> 1(1(3(4(5(0(x1)))))) 57.03/15.40 0(1(3(5(1(x1))))) -> 1(1(5(0(3(3(x1)))))) 57.03/15.40 0(1(5(2(0(x1))))) -> 5(0(3(1(0(2(x1)))))) 57.03/15.40 0(1(5(4(1(x1))))) -> 5(3(4(1(0(1(x1)))))) 57.03/15.40 0(1(5(4(4(x1))))) -> 4(5(2(1(4(0(x1)))))) 57.03/15.40 0(1(5(4(4(x1))))) -> 5(0(4(3(4(1(x1)))))) 57.03/15.40 0(3(1(0(4(x1))))) -> 0(0(3(1(2(4(x1)))))) 57.03/15.40 0(4(3(3(0(x1))))) -> 0(2(3(4(0(3(x1)))))) 57.03/15.40 0(4(5(2(0(x1))))) -> 0(2(2(5(0(4(x1)))))) 57.03/15.40 0(5(1(5(1(x1))))) -> 5(5(3(0(1(1(x1)))))) 57.03/15.40 3(0(1(5(4(x1))))) -> 0(5(3(4(1(2(x1)))))) 57.03/15.40 3(5(0(1(0(x1))))) -> 5(1(2(0(0(3(x1)))))) 57.03/15.40 3(5(4(0(0(x1))))) -> 5(0(3(0(4(4(x1)))))) 57.03/15.40 3(5(5(0(1(x1))))) -> 5(5(0(3(1(2(x1)))))) 57.03/15.40 57.03/15.40 Q is empty. 57.03/15.40 57.03/15.40 ---------------------------------------- 57.03/15.40 57.03/15.40 (1) QTRS Reverse (EQUIVALENT) 57.03/15.40 We applied the QTRS Reverse Processor [REVERSE]. 57.03/15.40 ---------------------------------------- 57.03/15.40 57.03/15.40 (2) 57.03/15.40 Obligation: 57.03/15.40 Q restricted rewrite system: 57.03/15.40 The TRS R consists of the following rules: 57.03/15.40 57.03/15.40 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.40 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.40 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.40 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.40 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.40 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.40 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.40 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.40 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.40 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.40 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.40 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.40 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.40 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.40 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.40 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.40 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.40 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.40 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.40 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.40 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.40 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.40 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.40 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.40 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.40 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.40 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.40 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.40 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.40 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.40 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.40 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.40 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.40 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.40 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.40 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.40 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.40 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.40 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.40 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.40 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.40 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.40 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.40 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.40 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.40 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.40 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.40 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.40 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.40 57.03/15.40 Q is empty. 57.03/15.40 57.03/15.40 ---------------------------------------- 57.03/15.40 57.03/15.40 (3) DependencyPairsProof (EQUIVALENT) 57.03/15.40 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 57.03/15.40 ---------------------------------------- 57.03/15.40 57.03/15.40 (4) 57.03/15.40 Obligation: 57.03/15.40 Q DP problem: 57.03/15.40 The TRS P consists of the following rules: 57.03/15.40 57.03/15.40 1^1(0(0(x1))) -> 1^1(2(0(0(x1)))) 57.03/15.40 1^1(0(0(x1))) -> 4^1(1(3(0(0(x1))))) 57.03/15.40 1^1(0(0(x1))) -> 1^1(3(0(0(x1)))) 57.03/15.40 0^1(1(0(x1))) -> 4^1(2(1(0(0(x1))))) 57.03/15.40 0^1(1(0(x1))) -> 1^1(0(0(x1))) 57.03/15.40 0^1(1(0(x1))) -> 0^1(0(x1)) 57.03/15.40 0^1(1(0(x1))) -> 0^1(0(2(1(2(x1))))) 57.03/15.40 0^1(1(0(x1))) -> 0^1(2(1(2(x1)))) 57.03/15.40 0^1(1(0(x1))) -> 1^1(2(x1)) 57.03/15.40 0^1(1(0(x1))) -> 4^1(2(2(1(0(0(x1)))))) 57.03/15.40 0^1(1(0(x1))) -> 4^1(1(4(2(0(0(x1)))))) 57.03/15.40 0^1(1(0(x1))) -> 1^1(4(2(0(0(x1))))) 57.03/15.40 0^1(1(0(x1))) -> 4^1(2(0(0(x1)))) 57.03/15.40 0^1(4(0(x1))) -> 0^1(0(2(3(4(3(x1)))))) 57.03/15.40 0^1(4(0(x1))) -> 0^1(2(3(4(3(x1))))) 57.03/15.40 0^1(4(0(x1))) -> 4^1(3(x1)) 57.03/15.40 0^1(1(0(0(x1)))) -> 4^1(1(0(0(0(x1))))) 57.03/15.40 0^1(1(0(0(x1)))) -> 1^1(0(0(0(x1)))) 57.03/15.40 0^1(1(0(0(x1)))) -> 0^1(0(0(x1))) 57.03/15.40 1^1(4(0(0(x1)))) -> 4^1(1(2(0(0(x1))))) 57.03/15.40 1^1(4(0(0(x1)))) -> 1^1(2(0(0(x1)))) 57.03/15.40 4^1(0(1(0(x1)))) -> 1^1(4(3(0(0(x1))))) 57.03/15.40 4^1(0(1(0(x1)))) -> 4^1(3(0(0(x1)))) 57.03/15.40 4^1(0(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.40 0^1(3(1(0(x1)))) -> 4^1(1(3(0(0(x1))))) 57.03/15.40 0^1(3(1(0(x1)))) -> 1^1(3(0(0(x1)))) 57.03/15.40 0^1(3(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.40 0^1(3(1(0(x1)))) -> 1^1(0(3(0(x1)))) 57.03/15.40 0^1(3(1(0(x1)))) -> 0^1(3(0(x1))) 57.03/15.40 0^1(3(1(0(x1)))) -> 1^1(0(0(x1))) 57.03/15.40 0^1(4(1(0(x1)))) -> 4^1(1(3(0(2(0(x1)))))) 57.03/15.40 0^1(4(1(0(x1)))) -> 1^1(3(0(2(0(x1))))) 57.03/15.40 0^1(4(1(0(x1)))) -> 0^1(2(0(x1))) 57.03/15.40 1^1(5(1(0(x1)))) -> 0^1(5(4(1(1(2(x1)))))) 57.03/15.40 1^1(5(1(0(x1)))) -> 4^1(1(1(2(x1)))) 57.03/15.40 1^1(5(1(0(x1)))) -> 1^1(1(2(x1))) 57.03/15.40 1^1(5(1(0(x1)))) -> 1^1(2(x1)) 57.03/15.40 4^1(5(1(0(x1)))) -> 0^1(5(2(4(2(1(x1)))))) 57.03/15.40 4^1(5(1(0(x1)))) -> 4^1(2(1(x1))) 57.03/15.40 4^1(5(1(0(x1)))) -> 1^1(x1) 57.03/15.40 4^1(5(1(0(x1)))) -> 0^1(1(4(x1))) 57.03/15.40 4^1(5(1(0(x1)))) -> 1^1(4(x1)) 57.03/15.40 4^1(5(1(0(x1)))) -> 4^1(x1) 57.03/15.40 4^1(5(1(0(x1)))) -> 4^1(1(4(2(0(5(x1)))))) 57.03/15.40 4^1(5(1(0(x1)))) -> 1^1(4(2(0(5(x1))))) 57.03/15.40 4^1(5(1(0(x1)))) -> 4^1(2(0(5(x1)))) 57.03/15.40 4^1(5(1(0(x1)))) -> 0^1(5(x1)) 57.03/15.40 1^1(0(3(0(x1)))) -> 1^1(3(0(0(x1)))) 57.03/15.40 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.40 0^1(1(3(0(x1)))) -> 4^1(1(3(0(0(x1))))) 57.03/15.40 0^1(1(3(0(x1)))) -> 1^1(3(0(0(x1)))) 57.03/15.40 0^1(1(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.40 1^1(0(4(0(x1)))) -> 4^1(1(2(0(0(x1))))) 57.03/15.40 1^1(0(4(0(x1)))) -> 1^1(2(0(0(x1)))) 57.03/15.40 1^1(0(4(0(x1)))) -> 0^1(0(x1)) 57.03/15.40 1^1(5(4(0(x1)))) -> 0^1(5(2(4(2(1(x1)))))) 57.03/15.40 1^1(5(4(0(x1)))) -> 4^1(2(1(x1))) 57.03/15.40 1^1(5(4(0(x1)))) -> 1^1(x1) 57.03/15.40 1^1(5(4(0(x1)))) -> 0^1(5(4(1(3(x1))))) 57.03/15.40 1^1(5(4(0(x1)))) -> 4^1(1(3(x1))) 57.03/15.40 1^1(5(4(0(x1)))) -> 1^1(3(x1)) 57.03/15.40 1^1(5(4(0(x1)))) -> 0^1(4(1(5(2(3(x1)))))) 57.03/15.40 1^1(5(4(0(x1)))) -> 4^1(1(5(2(3(x1))))) 57.03/15.40 1^1(5(4(0(x1)))) -> 1^1(5(2(3(x1)))) 57.03/15.40 1^1(5(4(0(x1)))) -> 4^1(1(5(0(3(5(x1)))))) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(5(0(3(5(x1))))) 57.03/15.41 1^1(5(4(0(x1)))) -> 0^1(3(5(x1))) 57.03/15.41 1^1(5(4(0(x1)))) -> 4^1(1(5(0(5(5(x1)))))) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(5(0(5(5(x1))))) 57.03/15.41 1^1(5(4(0(x1)))) -> 0^1(5(5(x1))) 57.03/15.41 4^1(5(4(0(x1)))) -> 4^1(0(4(4(2(5(x1)))))) 57.03/15.41 4^1(5(4(0(x1)))) -> 0^1(4(4(2(5(x1))))) 57.03/15.41 4^1(5(4(0(x1)))) -> 4^1(4(2(5(x1)))) 57.03/15.41 4^1(5(4(0(x1)))) -> 4^1(2(5(x1))) 57.03/15.41 0^1(1(5(0(x1)))) -> 1^1(5(0(0(x1)))) 57.03/15.41 0^1(1(5(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 1^1(0(5(3(x1)))) -> 1^1(2(0(3(x1)))) 57.03/15.41 1^1(0(5(3(x1)))) -> 0^1(3(x1)) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(0(x1))) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(x1) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(2(1(2(x1)))) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(2(x1)) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(2(2(1(3(x1))))) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(3(x1)) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(2(3(1(5(x1))))) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(x1)) 57.03/15.41 0^1(3(3(1(0(x1))))) -> 1^1(0(0(2(3(3(x1)))))) 57.03/15.41 0^1(3(3(1(0(x1))))) -> 0^1(0(2(3(3(x1))))) 57.03/15.41 0^1(3(3(1(0(x1))))) -> 0^1(2(3(3(x1)))) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 0^1(5(4(3(1(1(x1)))))) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 4^1(3(1(1(x1)))) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(1(x1)) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(x1) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 0^1(5(1(1(x1)))) 57.03/15.41 0^1(2(5(1(0(x1))))) -> 0^1(1(3(0(5(x1))))) 57.03/15.41 0^1(2(5(1(0(x1))))) -> 1^1(3(0(5(x1)))) 57.03/15.41 0^1(2(5(1(0(x1))))) -> 0^1(5(x1)) 57.03/15.41 1^1(4(5(1(0(x1))))) -> 1^1(0(1(4(3(5(x1)))))) 57.03/15.41 1^1(4(5(1(0(x1))))) -> 0^1(1(4(3(5(x1))))) 57.03/15.41 1^1(4(5(1(0(x1))))) -> 1^1(4(3(5(x1)))) 57.03/15.41 1^1(4(5(1(0(x1))))) -> 4^1(3(5(x1))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 0^1(4(1(2(5(4(x1)))))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(1(2(5(4(x1))))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 1^1(2(5(4(x1)))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(x1) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 1^1(4(3(4(0(5(x1)))))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(3(4(0(5(x1))))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(0(5(x1))) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 0^1(5(x1)) 57.03/15.41 4^1(0(1(3(0(x1))))) -> 4^1(2(1(3(0(0(x1)))))) 57.03/15.41 4^1(0(1(3(0(x1))))) -> 1^1(3(0(0(x1)))) 57.03/15.41 4^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 57.03/15.41 0^1(3(3(4(0(x1))))) -> 0^1(4(3(2(0(x1))))) 57.03/15.41 0^1(3(3(4(0(x1))))) -> 4^1(3(2(0(x1)))) 57.03/15.41 0^1(2(5(4(0(x1))))) -> 4^1(0(5(2(2(0(x1)))))) 57.03/15.41 0^1(2(5(4(0(x1))))) -> 0^1(5(2(2(0(x1))))) 57.03/15.41 1^1(5(1(5(0(x1))))) -> 1^1(1(0(3(5(5(x1)))))) 57.03/15.41 1^1(5(1(5(0(x1))))) -> 1^1(0(3(5(5(x1))))) 57.03/15.41 1^1(5(1(5(0(x1))))) -> 0^1(3(5(5(x1)))) 57.03/15.41 4^1(5(1(0(3(x1))))) -> 1^1(4(3(5(0(x1))))) 57.03/15.41 4^1(5(1(0(3(x1))))) -> 4^1(3(5(0(x1)))) 57.03/15.41 4^1(5(1(0(3(x1))))) -> 0^1(x1) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(2(1(5(x1)))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 1^1(5(x1)) 57.03/15.41 0^1(0(4(5(3(x1))))) -> 4^1(4(0(3(0(5(x1)))))) 57.03/15.41 0^1(0(4(5(3(x1))))) -> 4^1(0(3(0(5(x1))))) 57.03/15.41 0^1(0(4(5(3(x1))))) -> 0^1(3(0(5(x1)))) 57.03/15.41 0^1(0(4(5(3(x1))))) -> 0^1(5(x1)) 57.03/15.41 1^1(0(5(5(3(x1))))) -> 1^1(3(0(5(5(x1))))) 57.03/15.41 1^1(0(5(5(3(x1))))) -> 0^1(5(5(x1))) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (5) DependencyGraphProof (EQUIVALENT) 57.03/15.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 97 less nodes. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (6) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(x1))) -> 1^1(0(0(x1))) 57.03/15.41 1^1(0(4(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(x1))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 4^1(1(0(0(0(x1))))) 57.03/15.41 4^1(0(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 1^1(0(0(0(x1)))) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(x1) 57.03/15.41 1^1(0(5(3(x1)))) -> 0^1(3(x1)) 57.03/15.41 0^1(3(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 0^1(0(0(x1))) 57.03/15.41 0^1(3(1(0(x1)))) -> 1^1(0(3(0(x1)))) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(1(x1)) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(x1) 57.03/15.41 0^1(3(1(0(x1)))) -> 0^1(3(0(x1))) 57.03/15.41 0^1(3(1(0(x1)))) -> 1^1(0(0(x1))) 57.03/15.41 0^1(4(1(0(x1)))) -> 0^1(2(0(x1))) 57.03/15.41 0^1(2(5(1(0(x1))))) -> 0^1(1(3(0(5(x1))))) 57.03/15.41 0^1(1(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(5(0(x1)))) -> 1^1(5(0(0(x1)))) 57.03/15.41 0^1(1(5(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(0(x1))) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(x1) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(x1)) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(2(1(5(x1)))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 1^1(5(x1)) 57.03/15.41 4^1(5(1(0(x1)))) -> 1^1(x1) 57.03/15.41 4^1(5(1(0(x1)))) -> 0^1(1(4(x1))) 57.03/15.41 4^1(5(1(0(x1)))) -> 1^1(4(x1)) 57.03/15.41 4^1(5(1(0(x1)))) -> 4^1(x1) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(x1) 57.03/15.41 4^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 57.03/15.41 4^1(5(1(0(3(x1))))) -> 0^1(x1) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (7) QDPOrderProof (EQUIVALENT) 57.03/15.41 We use the reduction pair processor [LPAR04,JAR06]. 57.03/15.41 57.03/15.41 57.03/15.41 The following pairs can be oriented strictly and are deleted. 57.03/15.41 57.03/15.41 0^1(1(0(x1))) -> 1^1(0(0(x1))) 57.03/15.41 0^1(1(0(x1))) -> 0^1(0(x1)) 57.03/15.41 4^1(0(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 1^1(0(0(0(x1)))) 57.03/15.41 0^1(3(1(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 0^1(0(0(x1))) 57.03/15.41 0^1(3(1(0(x1)))) -> 1^1(0(3(0(x1)))) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(x1) 57.03/15.41 0^1(3(1(0(x1)))) -> 0^1(3(0(x1))) 57.03/15.41 0^1(3(1(0(x1)))) -> 1^1(0(0(x1))) 57.03/15.41 0^1(4(1(0(x1)))) -> 0^1(2(0(x1))) 57.03/15.41 0^1(1(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(5(0(x1)))) -> 1^1(5(0(0(x1)))) 57.03/15.41 0^1(1(5(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(0(x1))) 57.03/15.41 0^1(1(5(3(x1)))) -> 0^1(x1) 57.03/15.41 0^1(1(5(3(x1)))) -> 1^1(5(x1)) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 1^1(5(x1)) 57.03/15.41 4^1(5(1(0(x1)))) -> 1^1(x1) 57.03/15.41 4^1(5(1(0(x1)))) -> 1^1(4(x1)) 57.03/15.41 4^1(5(1(0(x1)))) -> 4^1(x1) 57.03/15.41 4^1(4(5(1(0(x1))))) -> 4^1(x1) 57.03/15.41 4^1(0(1(3(0(x1))))) -> 0^1(0(x1)) 57.03/15.41 4^1(5(1(0(3(x1))))) -> 0^1(x1) 57.03/15.41 The remaining pairs can at least be oriented weakly. 57.03/15.41 Used ordering: Polynomial interpretation [POLO]: 57.03/15.41 57.03/15.41 POL(0(x_1)) = x_1 57.03/15.41 POL(0^1(x_1)) = x_1 57.03/15.41 POL(1(x_1)) = 1 + x_1 57.03/15.41 POL(1^1(x_1)) = x_1 57.03/15.41 POL(2(x_1)) = x_1 57.03/15.41 POL(3(x_1)) = x_1 57.03/15.41 POL(4(x_1)) = x_1 57.03/15.41 POL(4^1(x_1)) = x_1 57.03/15.41 POL(5(x_1)) = x_1 57.03/15.41 57.03/15.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 57.03/15.41 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (8) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 1^1(0(3(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 1^1(0(4(0(x1)))) -> 0^1(0(x1)) 57.03/15.41 0^1(1(0(0(x1)))) -> 4^1(1(0(0(0(x1))))) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(x1) 57.03/15.41 1^1(0(5(3(x1)))) -> 0^1(3(x1)) 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(1(x1)) 57.03/15.41 0^1(2(5(1(0(x1))))) -> 0^1(1(3(0(5(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(2(1(5(x1)))) 57.03/15.41 4^1(5(1(0(x1)))) -> 0^1(1(4(x1))) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (9) DependencyGraphProof (EQUIVALENT) 57.03/15.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (10) 57.03/15.41 Complex Obligation (AND) 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (11) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 4^1(5(1(0(x1)))) -> 0^1(1(4(x1))) 57.03/15.41 0^1(1(0(0(x1)))) -> 4^1(1(0(0(0(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (12) QDPOrderProof (EQUIVALENT) 57.03/15.41 We use the reduction pair processor [LPAR04,JAR06]. 57.03/15.41 57.03/15.41 57.03/15.41 The following pairs can be oriented strictly and are deleted. 57.03/15.41 57.03/15.41 4^1(5(1(0(x1)))) -> 0^1(1(4(x1))) 57.03/15.41 The remaining pairs can at least be oriented weakly. 57.03/15.41 Used ordering: Polynomial interpretation [POLO]: 57.03/15.41 57.03/15.41 POL(0(x_1)) = 0 57.03/15.41 POL(0^1(x_1)) = 1 57.03/15.41 POL(1(x_1)) = 1 57.03/15.41 POL(2(x_1)) = 0 57.03/15.41 POL(3(x_1)) = 0 57.03/15.41 POL(4(x_1)) = 0 57.03/15.41 POL(4^1(x_1)) = x_1 57.03/15.41 POL(5(x_1)) = 1 + x_1 57.03/15.41 57.03/15.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (13) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 0^1(1(0(0(x1)))) -> 4^1(1(0(0(0(x1))))) 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (14) DependencyGraphProof (EQUIVALENT) 57.03/15.41 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (15) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (16) QDPOrderProof (EQUIVALENT) 57.03/15.41 We use the reduction pair processor [LPAR04,JAR06]. 57.03/15.41 57.03/15.41 57.03/15.41 The following pairs can be oriented strictly and are deleted. 57.03/15.41 57.03/15.41 0^1(1(0(5(3(x1))))) -> 0^1(0(2(1(5(x1))))) 57.03/15.41 The remaining pairs can at least be oriented weakly. 57.03/15.41 Used ordering: Polynomial interpretation [POLO]: 57.03/15.41 57.03/15.41 POL(0(x_1)) = 0 57.03/15.41 POL(0^1(x_1)) = x_1 57.03/15.41 POL(1(x_1)) = 1 57.03/15.41 POL(2(x_1)) = 0 57.03/15.41 POL(3(x_1)) = 0 57.03/15.41 POL(4(x_1)) = 0 57.03/15.41 POL(5(x_1)) = 0 57.03/15.41 57.03/15.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 57.03/15.41 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (17) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 P is empty. 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (18) PisEmptyProof (EQUIVALENT) 57.03/15.41 The TRS P is empty. Hence, there is no (P,Q,R) chain. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (19) 57.03/15.41 YES 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (20) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 The TRS P consists of the following rules: 57.03/15.41 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(1(x1)) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(x1) 57.03/15.41 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (21) QDPOrderProof (EQUIVALENT) 57.03/15.41 We use the reduction pair processor [LPAR04,JAR06]. 57.03/15.41 57.03/15.41 57.03/15.41 The following pairs can be oriented strictly and are deleted. 57.03/15.41 57.03/15.41 1^1(5(3(1(0(x1))))) -> 1^1(1(x1)) 57.03/15.41 1^1(5(4(0(x1)))) -> 1^1(x1) 57.03/15.41 The remaining pairs can at least be oriented weakly. 57.03/15.41 Used ordering: Polynomial interpretation [POLO]: 57.03/15.41 57.03/15.41 POL(0(x_1)) = 1 + x_1 57.03/15.41 POL(1(x_1)) = 1 + x_1 57.03/15.41 POL(1^1(x_1)) = x_1 57.03/15.41 POL(2(x_1)) = 0 57.03/15.41 POL(3(x_1)) = x_1 57.03/15.41 POL(4(x_1)) = x_1 57.03/15.41 POL(5(x_1)) = x_1 57.03/15.41 57.03/15.41 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 57.03/15.41 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (22) 57.03/15.41 Obligation: 57.03/15.41 Q DP problem: 57.03/15.41 P is empty. 57.03/15.41 The TRS R consists of the following rules: 57.03/15.41 57.03/15.41 1(0(0(x1))) -> 2(1(2(0(0(x1))))) 57.03/15.41 1(0(0(x1))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(1(0(0(x1))))) 57.03/15.41 0(1(0(x1))) -> 0(0(2(1(2(x1))))) 57.03/15.41 0(1(0(x1))) -> 4(2(2(1(0(0(x1)))))) 57.03/15.41 0(1(0(x1))) -> 4(1(4(2(0(0(x1)))))) 57.03/15.41 0(4(0(x1))) -> 0(0(2(3(4(3(x1)))))) 57.03/15.41 0(1(0(0(x1)))) -> 4(1(0(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(4(0(0(x1)))) -> 3(4(1(2(0(0(x1)))))) 57.03/15.41 4(0(1(0(x1)))) -> 2(1(4(3(0(0(x1)))))) 57.03/15.41 0(3(1(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 2(1(0(3(0(x1))))) 57.03/15.41 0(3(1(0(x1)))) -> 5(2(3(1(0(0(x1)))))) 57.03/15.41 0(4(1(0(x1)))) -> 4(1(3(0(2(0(x1)))))) 57.03/15.41 1(5(1(0(x1)))) -> 0(5(4(1(1(2(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 5(2(3(0(1(4(x1)))))) 57.03/15.41 4(5(1(0(x1)))) -> 4(1(4(2(0(5(x1)))))) 57.03/15.41 1(0(3(0(x1)))) -> 2(2(1(3(0(0(x1)))))) 57.03/15.41 0(1(3(0(x1)))) -> 4(1(3(0(0(x1))))) 57.03/15.41 1(0(4(0(x1)))) -> 4(1(2(0(0(x1))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(5(2(4(2(1(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 2(0(5(4(1(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 0(4(1(5(2(3(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(3(5(x1)))))) 57.03/15.41 1(5(4(0(x1)))) -> 4(1(5(0(5(5(x1)))))) 57.03/15.41 4(5(4(0(x1)))) -> 4(0(4(4(2(5(x1)))))) 57.03/15.41 0(1(5(0(x1)))) -> 2(1(5(0(0(x1))))) 57.03/15.41 1(0(5(3(x1)))) -> 5(2(1(2(0(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(3(1(5(0(x1))))) 57.03/15.41 0(1(5(3(x1)))) -> 3(5(0(2(1(2(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 5(0(2(2(1(3(x1)))))) 57.03/15.41 0(1(5(3(x1)))) -> 2(0(2(3(1(5(x1)))))) 57.03/15.41 0(3(3(1(0(x1))))) -> 1(0(0(2(3(3(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 0(5(4(3(1(1(x1)))))) 57.03/15.41 1(5(3(1(0(x1))))) -> 3(3(0(5(1(1(x1)))))) 57.03/15.41 0(2(5(1(0(x1))))) -> 2(0(1(3(0(5(x1)))))) 57.03/15.41 1(4(5(1(0(x1))))) -> 1(0(1(4(3(5(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 0(4(1(2(5(4(x1)))))) 57.03/15.41 4(4(5(1(0(x1))))) -> 1(4(3(4(0(5(x1)))))) 57.03/15.41 4(0(1(3(0(x1))))) -> 4(2(1(3(0(0(x1)))))) 57.03/15.41 0(3(3(4(0(x1))))) -> 3(0(4(3(2(0(x1)))))) 57.03/15.41 0(2(5(4(0(x1))))) -> 4(0(5(2(2(0(x1)))))) 57.03/15.41 1(5(1(5(0(x1))))) -> 1(1(0(3(5(5(x1)))))) 57.03/15.41 4(5(1(0(3(x1))))) -> 2(1(4(3(5(0(x1)))))) 57.03/15.41 0(1(0(5(3(x1))))) -> 3(0(0(2(1(5(x1)))))) 57.03/15.41 0(0(4(5(3(x1))))) -> 4(4(0(3(0(5(x1)))))) 57.03/15.41 1(0(5(5(3(x1))))) -> 2(1(3(0(5(5(x1)))))) 57.03/15.41 57.03/15.41 Q is empty. 57.03/15.41 We have to consider all minimal (P,Q,R)-chains. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (23) PisEmptyProof (EQUIVALENT) 57.03/15.41 The TRS P is empty. Hence, there is no (P,Q,R) chain. 57.03/15.41 ---------------------------------------- 57.03/15.41 57.03/15.41 (24) 57.03/15.41 YES 57.44/15.60 EOF