YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 246 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 7 ms] (6) QDP (7) QDPOrderProof [EQUIVALENT, 5662 ms] (8) QDP (9) QDPOrderProof [EQUIVALENT, 4775 ms] (10) QDP (11) QDPOrderProof [EQUIVALENT, 3114 ms] (12) QDP (13) QDPOrderProof [EQUIVALENT, 4204 ms] (14) QDP (15) QDPOrderProof [EQUIVALENT, 4002 ms] (16) QDP (17) QDPOrderProof [EQUIVALENT, 4690 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 3047 ms] (20) QDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 5124 ms] (24) QDP (25) QDPOrderProof [EQUIVALENT, 4830 ms] (26) QDP (27) QDPOrderProof [EQUIVALENT, 5414 ms] (28) QDP (29) QDPOrderProof [EQUIVALENT, 6839 ms] (30) QDP (31) QDPOrderProof [EQUIVALENT, 4652 ms] (32) QDP (33) QDPOrderProof [EQUIVALENT, 5741 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) AND (37) QDP (38) QDPOrderProof [EQUIVALENT, 5515 ms] (39) QDP (40) PisEmptyProof [EQUIVALENT, 0 ms] (41) YES (42) QDP (43) QDPOrderProof [EQUIVALENT, 1379 ms] (44) QDP (45) QDPOrderProof [EQUIVALENT, 1186 ms] (46) QDP (47) QDPOrderProof [EQUIVALENT, 21 ms] (48) QDP (49) PisEmptyProof [EQUIVALENT, 0 ms] (50) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(0(x1))) -> 2(0(0(1(x1)))) 0(1(0(x1))) -> 0(1(2(2(0(x1))))) 0(1(0(x1))) -> 2(0(0(3(1(x1))))) 0(1(0(x1))) -> 0(1(2(4(0(3(x1)))))) 0(1(0(x1))) -> 1(2(2(0(0(4(x1)))))) 0(1(0(x1))) -> 1(2(3(4(0(0(x1)))))) 0(1(0(x1))) -> 1(3(2(0(0(0(x1)))))) 0(1(0(x1))) -> 1(3(4(0(2(0(x1)))))) 0(1(0(x1))) -> 1(3(4(4(0(0(x1)))))) 0(1(0(x1))) -> 2(0(0(1(2(2(x1)))))) 0(1(0(x1))) -> 2(0(0(2(0(1(x1)))))) 0(1(0(x1))) -> 2(0(0(3(0(1(x1)))))) 0(1(0(x1))) -> 2(0(2(0(4(1(x1)))))) 0(1(0(x1))) -> 2(1(2(2(0(0(x1)))))) 0(1(0(x1))) -> 2(4(0(4(0(1(x1)))))) 0(1(0(x1))) -> 4(1(2(2(0(0(x1)))))) 0(1(1(0(x1)))) -> 2(2(0(0(1(1(x1)))))) 0(1(5(0(x1)))) -> 0(3(5(1(0(x1))))) 0(1(5(0(x1)))) -> 0(3(3(5(1(0(x1)))))) 0(5(5(0(x1)))) -> 2(2(0(0(5(5(x1)))))) 1(0(1(0(x1)))) -> 1(1(3(5(0(0(x1)))))) 1(3(0(1(x1)))) -> 1(1(0(3(4(1(x1)))))) 1(3(0(1(x1)))) -> 5(1(1(3(4(0(x1)))))) 3(0(1(0(x1)))) -> 0(1(3(2(0(4(x1)))))) 3(0(1(0(x1)))) -> 2(0(4(3(0(1(x1)))))) 3(0(1(0(x1)))) -> 2(2(3(0(0(1(x1)))))) 3(0(1(0(x1)))) -> 3(0(2(2(0(1(x1)))))) 3(1(0(0(x1)))) -> 1(0(3(4(0(x1))))) 3(1(0(0(x1)))) -> 2(0(3(1(0(x1))))) 3(1(0(0(x1)))) -> 3(0(0(5(1(x1))))) 3(1(0(0(x1)))) -> 1(2(0(3(0(2(x1)))))) 3(1(0(0(x1)))) -> 3(1(3(0(2(0(x1)))))) 3(1(5(0(x1)))) -> 1(1(3(5(0(x1))))) 3(1(5(0(x1)))) -> 0(2(5(1(3(5(x1)))))) 3(1(5(0(x1)))) -> 0(3(1(3(4(5(x1)))))) 3(1(5(0(x1)))) -> 1(5(2(3(5(0(x1)))))) 3(1(5(0(x1)))) -> 3(1(4(0(2(5(x1)))))) 3(1(5(0(x1)))) -> 4(0(5(1(3(5(x1)))))) 0(0(2(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 0(1(2(0(1(x1))))) -> 0(0(1(3(2(1(x1)))))) 0(1(2(0(1(x1))))) -> 0(2(0(3(1(1(x1)))))) 1(3(0(2(1(x1))))) -> 0(2(1(1(3(2(x1)))))) 1(3(0(2(1(x1))))) -> 0(3(2(2(1(1(x1)))))) 1(3(0(5(5(x1))))) -> 0(5(4(3(5(1(x1)))))) 1(3(1(5(0(x1))))) -> 1(3(1(5(2(0(x1)))))) 3(0(5(0(0(x1))))) -> 3(0(0(3(5(0(x1)))))) 3(0(5(5(0(x1))))) -> 5(0(2(3(5(0(x1)))))) 3(1(4(0(0(x1))))) -> 1(0(0(3(4(0(x1)))))) 3(1(4(5(0(x1))))) -> 0(5(1(3(5(4(x1)))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(0(0(2(x1)))) 0^1(1(0(x1))) -> 0^1(0(2(x1))) 0^1(1(0(x1))) -> 0^1(2(x1)) 0^1(1(0(x1))) -> 0^1(2(2(1(0(x1))))) 0^1(1(0(x1))) -> 1^1(3(0(0(2(x1))))) 0^1(1(0(x1))) -> 0^1(4(2(1(0(x1))))) 0^1(1(0(x1))) -> 0^1(0(2(2(1(x1))))) 0^1(1(0(x1))) -> 0^1(2(2(1(x1)))) 0^1(1(0(x1))) -> 1^1(x1) 0^1(1(0(x1))) -> 0^1(0(4(3(2(1(x1)))))) 0^1(1(0(x1))) -> 0^1(4(3(2(1(x1))))) 0^1(1(0(x1))) -> 0^1(0(0(2(3(1(x1)))))) 0^1(1(0(x1))) -> 0^1(0(2(3(1(x1))))) 0^1(1(0(x1))) -> 0^1(2(3(1(x1)))) 0^1(1(0(x1))) -> 0^1(2(0(4(3(1(x1)))))) 0^1(1(0(x1))) -> 0^1(4(3(1(x1)))) 0^1(1(0(x1))) -> 0^1(0(4(4(3(1(x1)))))) 0^1(1(0(x1))) -> 0^1(4(4(3(1(x1))))) 0^1(1(0(x1))) -> 1^1(0(2(0(0(2(x1)))))) 0^1(1(0(x1))) -> 0^1(2(0(0(2(x1))))) 0^1(1(0(x1))) -> 1^1(0(3(0(0(2(x1)))))) 0^1(1(0(x1))) -> 0^1(3(0(0(2(x1))))) 0^1(1(0(x1))) -> 1^1(4(0(2(0(2(x1)))))) 0^1(1(0(x1))) -> 0^1(2(0(2(x1)))) 0^1(1(0(x1))) -> 0^1(0(2(2(1(2(x1)))))) 0^1(1(0(x1))) -> 0^1(2(2(1(2(x1))))) 0^1(1(0(x1))) -> 1^1(2(x1)) 0^1(1(0(x1))) -> 1^1(0(4(0(4(2(x1)))))) 0^1(1(0(x1))) -> 0^1(4(0(4(2(x1))))) 0^1(1(0(x1))) -> 0^1(4(2(x1))) 0^1(1(0(x1))) -> 0^1(0(2(2(1(4(x1)))))) 0^1(1(0(x1))) -> 0^1(2(2(1(4(x1))))) 0^1(1(0(x1))) -> 1^1(4(x1)) 0^1(1(1(0(x1)))) -> 1^1(1(0(0(2(2(x1)))))) 0^1(1(1(0(x1)))) -> 1^1(0(0(2(2(x1))))) 0^1(1(1(0(x1)))) -> 0^1(0(2(2(x1)))) 0^1(1(1(0(x1)))) -> 0^1(2(2(x1))) 0^1(5(1(0(x1)))) -> 0^1(1(5(3(0(x1))))) 0^1(5(1(0(x1)))) -> 1^1(5(3(0(x1)))) 0^1(5(1(0(x1)))) -> 5^1(3(0(x1))) 0^1(5(1(0(x1)))) -> 0^1(1(5(3(3(0(x1)))))) 0^1(5(1(0(x1)))) -> 1^1(5(3(3(0(x1))))) 0^1(5(1(0(x1)))) -> 5^1(3(3(0(x1)))) 0^1(5(5(0(x1)))) -> 5^1(5(0(0(2(2(x1)))))) 0^1(5(5(0(x1)))) -> 5^1(0(0(2(2(x1))))) 0^1(5(5(0(x1)))) -> 0^1(0(2(2(x1)))) 0^1(5(5(0(x1)))) -> 0^1(2(2(x1))) 0^1(1(0(1(x1)))) -> 0^1(0(5(3(1(1(x1)))))) 0^1(1(0(1(x1)))) -> 0^1(5(3(1(1(x1))))) 0^1(1(0(1(x1)))) -> 5^1(3(1(1(x1)))) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 1^1(4(3(0(1(1(x1)))))) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 0^1(4(3(1(1(5(x1)))))) 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 0^1(1(0(3(x1)))) -> 0^1(2(3(1(0(x1))))) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(3(4(0(2(x1)))))) 0^1(1(0(3(x1)))) -> 0^1(3(4(0(2(x1))))) 0^1(1(0(3(x1)))) -> 0^1(2(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(0(3(2(2(x1)))))) 0^1(1(0(3(x1)))) -> 0^1(0(3(2(2(x1))))) 0^1(1(0(3(x1)))) -> 0^1(3(2(2(x1)))) 0^1(1(0(3(x1)))) -> 1^1(0(2(2(0(3(x1)))))) 0^1(1(0(3(x1)))) -> 0^1(2(2(0(3(x1))))) 0^1(0(1(3(x1)))) -> 0^1(4(3(0(1(x1))))) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(3(0(2(x1))))) 0^1(0(1(3(x1)))) -> 1^1(3(0(2(x1)))) 0^1(0(1(3(x1)))) -> 0^1(2(x1)) 0^1(0(1(3(x1)))) -> 1^1(5(0(0(3(x1))))) 0^1(0(1(3(x1)))) -> 5^1(0(0(3(x1)))) 0^1(0(1(3(x1)))) -> 0^1(0(3(x1))) 0^1(0(1(3(x1)))) -> 0^1(3(x1)) 0^1(0(1(3(x1)))) -> 0^1(3(0(2(1(x1))))) 0^1(0(1(3(x1)))) -> 0^1(2(1(x1))) 0^1(0(1(3(x1)))) -> 0^1(2(0(3(1(3(x1)))))) 0^1(0(1(3(x1)))) -> 0^1(3(1(3(x1)))) 0^1(5(1(3(x1)))) -> 0^1(5(3(1(1(x1))))) 0^1(5(1(3(x1)))) -> 5^1(3(1(1(x1)))) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 5^1(3(1(5(2(0(x1)))))) 0^1(5(1(3(x1)))) -> 1^1(5(2(0(x1)))) 0^1(5(1(3(x1)))) -> 5^1(2(0(x1))) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 5^1(4(3(1(3(0(x1)))))) 0^1(5(1(3(x1)))) -> 1^1(3(0(x1))) 0^1(5(1(3(x1)))) -> 0^1(5(3(2(5(1(x1)))))) 0^1(5(1(3(x1)))) -> 5^1(3(2(5(1(x1))))) 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 0^1(5(1(3(x1)))) -> 5^1(2(0(4(1(3(x1)))))) 0^1(5(1(3(x1)))) -> 0^1(4(1(3(x1)))) 0^1(5(1(3(x1)))) -> 5^1(3(1(5(0(4(x1)))))) 0^1(5(1(3(x1)))) -> 1^1(5(0(4(x1)))) 0^1(5(1(3(x1)))) -> 5^1(0(4(x1))) 0^1(5(1(3(x1)))) -> 0^1(4(x1)) 0^1(1(2(0(0(x1))))) -> 1^1(3(0(0(0(2(x1)))))) 0^1(1(2(0(0(x1))))) -> 0^1(0(0(2(x1)))) 0^1(1(2(0(0(x1))))) -> 0^1(0(2(x1))) 0^1(1(2(0(0(x1))))) -> 0^1(2(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(2(3(1(0(0(x1)))))) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(1(3(0(2(0(x1)))))) 1^1(0(2(1(0(x1))))) -> 1^1(3(0(2(0(x1))))) 1^1(0(2(1(0(x1))))) -> 0^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(2(3(0(x1)))))) 1^1(2(0(3(1(x1))))) -> 1^1(2(2(3(0(x1))))) 5^1(5(0(3(1(x1))))) -> 1^1(5(3(4(5(0(x1)))))) 5^1(5(0(3(1(x1))))) -> 5^1(3(4(5(0(x1))))) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(5(1(3(1(x1))))) -> 0^1(2(5(1(3(1(x1)))))) 0^1(0(5(0(3(x1))))) -> 0^1(5(3(0(0(3(x1)))))) 0^1(0(5(0(3(x1))))) -> 5^1(3(0(0(3(x1))))) 0^1(0(5(0(3(x1))))) -> 0^1(0(3(x1))) 0^1(5(5(0(3(x1))))) -> 0^1(5(3(2(0(5(x1)))))) 0^1(5(5(0(3(x1))))) -> 5^1(3(2(0(5(x1))))) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(4(3(0(0(1(x1)))))) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(3(1(5(0(x1))))) 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 106 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [1A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(3(1(x1)))) -> 1^1(5(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [1A], [-I]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[1A], [-I], [0A]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(1(3(x1)))) -> 5^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[1A]] + [[0A, -I, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(1^1(x_1)) = [[1A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(1(0(1(x1)))) -> 1^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(0(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, -I], [0A, -I, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 5^1(5(0(3(1(x1))))) -> 0^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 0A], [1A, 1A, 1A], [0A, 0A, -I]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(3(1(x1)))) -> 5^1(x1) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(5(5(0(3(x1))))) -> 5^1(x1) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[1A], [-I], [-I]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 0^1(1(0(3(x1)))) -> 0^1(x1) 0^1(5(1(3(x1)))) -> 0^1(x1) 0^1(5(4(1(3(x1))))) -> 0^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [1A], [-I]] + [[0A, -I, -I], [-I, 0A, -I], [1A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [-I], [1A]] + [[-I, -I, -I], [-I, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(1(3(x1)))) -> 1^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(3(1(x1)))) -> 1^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[-I], [0A], [1A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, -I]] * x_1 >>> <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 >>> <<< POL(1(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(1^1(x_1)) = [[1A]] + [[0A, 0A, 1A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(3(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 1^1(2(0(3(1(x1))))) -> 0^1(x1) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(2(0(3(1(x1))))) -> 0^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(0^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(1(x_1)) = [[1A], [1A], [0A]] + [[-I, 1A, -I], [-I, 1A, 0A], [-I, 0A, 1A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 >>> <<< POL(1^1(x_1)) = [[1A]] + [[-I, 0A, 0A]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(1(0(x1))) -> 1^1(x1) 0^1(1(0(x1))) -> 1^1(2(x1)) 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(1(3(x1)))) -> 1^1(x1) 0^1(5(1(3(x1)))) -> 1^1(x1) 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 1^1(x1) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. ---------------------------------------- (36) Complex Obligation (AND) ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 >>> <<< POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(0(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 >>> <<< POL(3(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 >>> <<< POL(1(x_1)) = [[1A], [1A], [-I]] + [[1A, 0A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> <<< POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] * x_1 >>> <<< POL(5(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (39) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (40) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (41) YES ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = max{0, 2x_1 - 2} POL( 5_1(x_1) ) = x_1 + 1 POL( 0_1(x_1) ) = x_1 + 1 POL( 3_1(x_1) ) = max{0, x_1 - 2} POL( 1_1(x_1) ) = 2 POL( 4_1(x_1) ) = max{0, 2x_1 - 2} POL( 2_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = 2x_1 POL( 5_1(x_1) ) = x_1 + 1 POL( 0_1(x_1) ) = x_1 + 1 POL( 3_1(x_1) ) = max{0, x_1 - 2} POL( 1_1(x_1) ) = 2 POL( 4_1(x_1) ) = max{0, 2x_1 - 1} POL( 2_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 0 POL(2(x_1)) = 0 POL(3(x_1)) = x_1 POL(4(x_1)) = 0 POL(5(x_1)) = 1 + x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) ---------------------------------------- (48) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(1(0(x1))) -> 1(0(0(2(x1)))) 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (50) YES