975.29/249.14 YES 996.90/254.61 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 996.90/254.61 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 996.90/254.61 996.90/254.61 996.90/254.61 Termination w.r.t. Q of the given QTRS could be proven: 996.90/254.61 996.90/254.61 (0) QTRS 996.90/254.61 (1) QTRS Reverse [EQUIVALENT, 0 ms] 996.90/254.61 (2) QTRS 996.90/254.61 (3) DependencyPairsProof [EQUIVALENT, 428 ms] 996.90/254.61 (4) QDP 996.90/254.61 (5) DependencyGraphProof [EQUIVALENT, 6 ms] 996.90/254.61 (6) QDP 996.90/254.61 (7) QDPOrderProof [EQUIVALENT, 5004 ms] 996.90/254.61 (8) QDP 996.90/254.61 (9) QDPOrderProof [EQUIVALENT, 4727 ms] 996.90/254.61 (10) QDP 996.90/254.61 (11) QDPOrderProof [EQUIVALENT, 3155 ms] 996.90/254.61 (12) QDP 996.90/254.61 (13) QDPOrderProof [EQUIVALENT, 4258 ms] 996.90/254.61 (14) QDP 996.90/254.61 (15) QDPOrderProof [EQUIVALENT, 3682 ms] 996.90/254.61 (16) QDP 996.90/254.61 (17) QDPOrderProof [EQUIVALENT, 4028 ms] 996.90/254.61 (18) QDP 996.90/254.61 (19) QDPOrderProof [EQUIVALENT, 7318 ms] 996.90/254.61 (20) QDP 996.90/254.61 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 996.90/254.61 (22) QDP 996.90/254.61 (23) QDPOrderProof [EQUIVALENT, 4080 ms] 996.90/254.61 (24) QDP 996.90/254.61 (25) QDPOrderProof [EQUIVALENT, 5395 ms] 996.90/254.61 (26) QDP 996.90/254.61 (27) QDPOrderProof [EQUIVALENT, 5450 ms] 996.90/254.61 (28) QDP 996.90/254.61 (29) QDPOrderProof [EQUIVALENT, 6611 ms] 996.90/254.61 (30) QDP 996.90/254.61 (31) QDPOrderProof [EQUIVALENT, 7997 ms] 996.90/254.61 (32) QDP 996.90/254.61 (33) QDPOrderProof [EQUIVALENT, 5263 ms] 996.90/254.61 (34) QDP 996.90/254.61 (35) DependencyGraphProof [EQUIVALENT, 0 ms] 996.90/254.61 (36) AND 996.90/254.61 (37) QDP 996.90/254.61 (38) QDPOrderProof [EQUIVALENT, 5018 ms] 996.90/254.61 (39) QDP 996.90/254.61 (40) PisEmptyProof [EQUIVALENT, 0 ms] 996.90/254.61 (41) YES 996.90/254.61 (42) QDP 996.90/254.61 (43) QDPOrderProof [EQUIVALENT, 1368 ms] 996.90/254.61 (44) QDP 996.90/254.61 (45) QDPOrderProof [EQUIVALENT, 1260 ms] 996.90/254.61 (46) QDP 996.90/254.61 (47) QDPOrderProof [EQUIVALENT, 21 ms] 996.90/254.61 (48) QDP 996.90/254.61 (49) PisEmptyProof [EQUIVALENT, 0 ms] 996.90/254.61 (50) YES 996.90/254.61 996.90/254.61 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (0) 996.90/254.61 Obligation: 996.90/254.61 Q restricted rewrite system: 996.90/254.61 The TRS R consists of the following rules: 996.90/254.61 996.90/254.61 0(1(0(x1))) -> 2(0(0(1(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(1(2(2(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 2(0(0(3(1(x1))))) 996.90/254.61 0(1(0(x1))) -> 0(1(2(4(0(3(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(2(2(0(0(4(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(2(3(4(0(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(3(2(0(0(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(3(4(0(2(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(3(4(4(0(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(0(0(1(2(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(0(0(2(0(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(0(0(3(0(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(0(2(0(4(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(1(2(2(0(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(4(0(4(0(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(1(2(2(0(0(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 2(2(0(0(1(1(x1)))))) 996.90/254.61 0(1(5(0(x1)))) -> 0(3(5(1(0(x1))))) 996.90/254.61 0(1(5(0(x1)))) -> 0(3(3(5(1(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 2(2(0(0(5(5(x1)))))) 996.90/254.61 1(0(1(0(x1)))) -> 1(1(3(5(0(0(x1)))))) 996.90/254.61 1(3(0(1(x1)))) -> 1(1(0(3(4(1(x1)))))) 996.90/254.61 1(3(0(1(x1)))) -> 5(1(1(3(4(0(x1)))))) 996.90/254.61 3(0(1(0(x1)))) -> 0(1(3(2(0(4(x1)))))) 996.90/254.61 3(0(1(0(x1)))) -> 2(0(4(3(0(1(x1)))))) 996.90/254.61 3(0(1(0(x1)))) -> 2(2(3(0(0(1(x1)))))) 996.90/254.61 3(0(1(0(x1)))) -> 3(0(2(2(0(1(x1)))))) 996.90/254.61 3(1(0(0(x1)))) -> 1(0(3(4(0(x1))))) 996.90/254.61 3(1(0(0(x1)))) -> 2(0(3(1(0(x1))))) 996.90/254.61 3(1(0(0(x1)))) -> 3(0(0(5(1(x1))))) 996.90/254.61 3(1(0(0(x1)))) -> 1(2(0(3(0(2(x1)))))) 996.90/254.61 3(1(0(0(x1)))) -> 3(1(3(0(2(0(x1)))))) 996.90/254.61 3(1(5(0(x1)))) -> 1(1(3(5(0(x1))))) 996.90/254.61 3(1(5(0(x1)))) -> 0(2(5(1(3(5(x1)))))) 996.90/254.61 3(1(5(0(x1)))) -> 0(3(1(3(4(5(x1)))))) 996.90/254.61 3(1(5(0(x1)))) -> 1(5(2(3(5(0(x1)))))) 996.90/254.61 3(1(5(0(x1)))) -> 3(1(4(0(2(5(x1)))))) 996.90/254.61 3(1(5(0(x1)))) -> 4(0(5(1(3(5(x1)))))) 996.90/254.61 0(0(2(1(0(x1))))) -> 2(0(0(0(3(1(x1)))))) 996.90/254.61 0(1(2(0(1(x1))))) -> 0(0(1(3(2(1(x1)))))) 996.90/254.61 0(1(2(0(1(x1))))) -> 0(2(0(3(1(1(x1)))))) 996.90/254.61 1(3(0(2(1(x1))))) -> 0(2(1(1(3(2(x1)))))) 996.90/254.61 1(3(0(2(1(x1))))) -> 0(3(2(2(1(1(x1)))))) 996.90/254.61 1(3(0(5(5(x1))))) -> 0(5(4(3(5(1(x1)))))) 996.90/254.61 1(3(1(5(0(x1))))) -> 1(3(1(5(2(0(x1)))))) 996.90/254.61 3(0(5(0(0(x1))))) -> 3(0(0(3(5(0(x1)))))) 996.90/254.61 3(0(5(5(0(x1))))) -> 5(0(2(3(5(0(x1)))))) 996.90/254.61 3(1(4(0(0(x1))))) -> 1(0(0(3(4(0(x1)))))) 996.90/254.61 3(1(4(5(0(x1))))) -> 0(5(1(3(5(4(x1)))))) 996.90/254.61 996.90/254.61 Q is empty. 996.90/254.61 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (1) QTRS Reverse (EQUIVALENT) 996.90/254.61 We applied the QTRS Reverse Processor [REVERSE]. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (2) 996.90/254.61 Obligation: 996.90/254.61 Q restricted rewrite system: 996.90/254.61 The TRS R consists of the following rules: 996.90/254.61 996.90/254.61 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.61 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.61 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.61 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.61 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.61 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.61 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.61 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.61 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.61 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.61 996.90/254.61 Q is empty. 996.90/254.61 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (3) DependencyPairsProof (EQUIVALENT) 996.90/254.61 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (4) 996.90/254.61 Obligation: 996.90/254.61 Q DP problem: 996.90/254.61 The TRS P consists of the following rules: 996.90/254.61 996.90/254.61 0^1(1(0(x1))) -> 1^1(0(0(2(x1)))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(2(x1))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(x1)) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(2(1(0(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(3(0(0(2(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(2(1(0(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(2(2(1(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(2(1(x1)))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(4(3(2(1(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(3(2(1(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(0(2(3(1(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(2(3(1(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(3(1(x1)))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(0(4(3(1(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(3(1(x1)))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(4(4(3(1(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(4(3(1(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(0(2(0(0(2(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(0(0(2(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(0(3(0(0(2(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(3(0(0(2(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(4(0(2(0(2(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(0(2(x1)))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(2(2(1(2(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(2(1(2(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.61 0^1(1(0(x1))) -> 1^1(0(4(0(4(2(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(0(4(2(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(4(2(x1))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(0(2(2(1(4(x1)))))) 996.90/254.61 0^1(1(0(x1))) -> 0^1(2(2(1(4(x1))))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(4(x1)) 996.90/254.61 0^1(1(1(0(x1)))) -> 1^1(1(0(0(2(2(x1)))))) 996.90/254.61 0^1(1(1(0(x1)))) -> 1^1(0(0(2(2(x1))))) 996.90/254.61 0^1(1(1(0(x1)))) -> 0^1(0(2(2(x1)))) 996.90/254.61 0^1(1(1(0(x1)))) -> 0^1(2(2(x1))) 996.90/254.61 0^1(5(1(0(x1)))) -> 0^1(1(5(3(0(x1))))) 996.90/254.61 0^1(5(1(0(x1)))) -> 1^1(5(3(0(x1)))) 996.90/254.61 0^1(5(1(0(x1)))) -> 5^1(3(0(x1))) 996.90/254.61 0^1(5(1(0(x1)))) -> 0^1(1(5(3(3(0(x1)))))) 996.90/254.61 0^1(5(1(0(x1)))) -> 1^1(5(3(3(0(x1))))) 996.90/254.61 0^1(5(1(0(x1)))) -> 5^1(3(3(0(x1)))) 996.90/254.61 0^1(5(5(0(x1)))) -> 5^1(5(0(0(2(2(x1)))))) 996.90/254.61 0^1(5(5(0(x1)))) -> 5^1(0(0(2(2(x1))))) 996.90/254.61 0^1(5(5(0(x1)))) -> 0^1(0(2(2(x1)))) 996.90/254.61 0^1(5(5(0(x1)))) -> 0^1(2(2(x1))) 996.90/254.61 0^1(1(0(1(x1)))) -> 0^1(0(5(3(1(1(x1)))))) 996.90/254.61 0^1(1(0(1(x1)))) -> 0^1(5(3(1(1(x1))))) 996.90/254.61 0^1(1(0(1(x1)))) -> 5^1(3(1(1(x1)))) 996.90/254.61 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(4(3(0(1(1(x1)))))) 996.90/254.61 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 0^1(4(3(1(1(5(x1)))))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(2(3(1(0(x1))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(3(4(0(2(x1)))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(3(4(0(2(x1))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(2(x1)) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(0(3(2(2(x1)))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(0(3(2(2(x1))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(3(2(2(x1)))) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(2(2(0(3(x1)))))) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(2(2(0(3(x1))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(4(3(0(1(x1))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(1(3(0(2(x1))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 1^1(3(0(2(x1)))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(2(x1)) 996.90/254.61 0^1(0(1(3(x1)))) -> 1^1(5(0(0(3(x1))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 5^1(0(0(3(x1)))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(0(3(x1))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(3(x1)) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(3(0(2(1(x1))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(2(1(x1))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(2(0(3(1(3(x1)))))) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(3(1(3(x1)))) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(5(3(1(1(x1))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(3(1(1(x1)))) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(3(1(5(2(0(x1)))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(5(2(0(x1)))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(2(0(x1))) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(4(3(1(3(0(x1)))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(3(0(x1))) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(5(3(2(5(1(x1)))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(3(2(5(1(x1))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(2(0(4(1(3(x1)))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(4(1(3(x1)))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(3(1(5(0(4(x1)))))) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(5(0(4(x1)))) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(0(4(x1))) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(4(x1)) 996.90/254.61 0^1(1(2(0(0(x1))))) -> 1^1(3(0(0(0(2(x1)))))) 996.90/254.61 0^1(1(2(0(0(x1))))) -> 0^1(0(0(2(x1)))) 996.90/254.61 0^1(1(2(0(0(x1))))) -> 0^1(0(2(x1))) 996.90/254.61 0^1(1(2(0(0(x1))))) -> 0^1(2(x1)) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(2(3(1(0(0(x1)))))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(1(3(0(2(0(x1)))))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(3(0(2(0(x1))))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 0^1(2(0(x1))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(1(2(2(3(0(x1)))))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(2(2(3(0(x1))))) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 1^1(5(3(4(5(0(x1)))))) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 5^1(3(4(5(0(x1))))) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 0^1(5(1(3(1(x1))))) -> 0^1(2(5(1(3(1(x1)))))) 996.90/254.61 0^1(0(5(0(3(x1))))) -> 0^1(5(3(0(0(3(x1)))))) 996.90/254.61 0^1(0(5(0(3(x1))))) -> 5^1(3(0(0(3(x1))))) 996.90/254.61 0^1(0(5(0(3(x1))))) -> 0^1(0(3(x1))) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 0^1(5(3(2(0(5(x1)))))) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 5^1(3(2(0(5(x1))))) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(4(3(0(0(1(x1)))))) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 5^1(3(1(5(0(x1))))) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.61 996.90/254.61 The TRS R consists of the following rules: 996.90/254.61 996.90/254.61 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.61 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.61 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.61 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.61 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.61 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.61 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.61 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.61 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.61 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.61 996.90/254.61 Q is empty. 996.90/254.61 We have to consider all minimal (P,Q,R)-chains. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (5) DependencyGraphProof (EQUIVALENT) 996.90/254.61 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 106 less nodes. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (6) 996.90/254.61 Obligation: 996.90/254.61 Q DP problem: 996.90/254.61 The TRS P consists of the following rules: 996.90/254.61 996.90/254.61 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.61 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.61 996.90/254.61 The TRS R consists of the following rules: 996.90/254.61 996.90/254.61 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.61 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.61 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.61 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.61 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.61 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.61 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.61 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.61 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.61 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.61 996.90/254.61 Q is empty. 996.90/254.61 We have to consider all minimal (P,Q,R)-chains. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (7) QDPOrderProof (EQUIVALENT) 996.90/254.61 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.61 996.90/254.61 996.90/254.61 The following pairs can be oriented strictly and are deleted. 996.90/254.61 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(5(x1))) 996.90/254.61 The remaining pairs can at least be oriented weakly. 996.90/254.61 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(0^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, -I, -I]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(0(x_1)) = [[1A], [1A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, 0A, 0A], [0A, -I, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(5^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(4(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 996.90/254.61 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.61 996.90/254.61 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.61 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.61 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.61 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.61 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.61 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.61 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.61 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.61 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.61 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.61 996.90/254.61 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (8) 996.90/254.61 Obligation: 996.90/254.61 Q DP problem: 996.90/254.61 The TRS P consists of the following rules: 996.90/254.61 996.90/254.61 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.61 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.61 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 996.90/254.61 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.61 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.61 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.61 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.61 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.61 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.61 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.61 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.61 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.61 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.61 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.61 996.90/254.61 The TRS R consists of the following rules: 996.90/254.61 996.90/254.61 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.61 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.61 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.61 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.61 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.61 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.61 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.61 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.61 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.61 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.61 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.61 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.61 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.61 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.61 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.61 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.61 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.61 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.61 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.61 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.61 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.61 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.61 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.61 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.61 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.61 996.90/254.61 Q is empty. 996.90/254.61 We have to consider all minimal (P,Q,R)-chains. 996.90/254.61 ---------------------------------------- 996.90/254.61 996.90/254.61 (9) QDPOrderProof (EQUIVALENT) 996.90/254.61 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.61 996.90/254.61 996.90/254.61 The following pairs can be oriented strictly and are deleted. 996.90/254.61 996.90/254.61 0^1(5(4(1(3(x1))))) -> 1^1(5(0(x1))) 996.90/254.61 The remaining pairs can at least be oriented weakly. 996.90/254.61 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.61 >>> 996.90/254.61 996.90/254.61 <<< 996.90/254.61 POL(0(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(1^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(3(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(5(x_1)) = [[0A], [0A], [1A]] + [[-I, -I, -I], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(4(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, -I, -I]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 996.90/254.62 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.62 996.90/254.62 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.62 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.62 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.62 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.62 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.62 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.62 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.62 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.62 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.62 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.62 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.62 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.62 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.62 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.62 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.62 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.62 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.62 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.62 996.90/254.62 996.90/254.62 ---------------------------------------- 996.90/254.62 996.90/254.62 (10) 996.90/254.62 Obligation: 996.90/254.62 Q DP problem: 996.90/254.62 The TRS P consists of the following rules: 996.90/254.62 996.90/254.62 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.62 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.62 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.62 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.62 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.62 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 996.90/254.62 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.62 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.62 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.62 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.62 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.62 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.62 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.62 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.62 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.62 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.62 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.62 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.62 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.62 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.62 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.62 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.62 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.62 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.62 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.62 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.62 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.62 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.62 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.62 996.90/254.62 The TRS R consists of the following rules: 996.90/254.62 996.90/254.62 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.62 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.62 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.62 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.62 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.62 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.62 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.62 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.62 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.62 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.62 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.62 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.62 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.62 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.62 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.62 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.62 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.62 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.62 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.62 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.62 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.62 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.62 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.62 996.90/254.62 Q is empty. 996.90/254.62 We have to consider all minimal (P,Q,R)-chains. 996.90/254.62 ---------------------------------------- 996.90/254.62 996.90/254.62 (11) QDPOrderProof (EQUIVALENT) 996.90/254.62 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.62 996.90/254.62 996.90/254.62 The following pairs can be oriented strictly and are deleted. 996.90/254.62 996.90/254.62 1^1(0(3(1(x1)))) -> 1^1(5(x1)) 996.90/254.62 The remaining pairs can at least be oriented weakly. 996.90/254.62 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(0(x_1)) = [[1A], [1A], [-I]] + [[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(2(x_1)) = [[0A], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [-I, -I, -I], [-I, -I, -I]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 <<< 996.90/254.62 POL(4(x_1)) = [[1A], [-I], [0A]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 996.90/254.62 >>> 996.90/254.62 996.90/254.62 996.90/254.62 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.62 996.90/254.62 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.62 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.62 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.62 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.62 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.62 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.62 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.62 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.62 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.62 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (12) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (13) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 0^1(5(1(3(x1)))) -> 5^1(1(x1)) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[1A]] + [[0A, -I, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, 0A], [-I, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[1A], [-I], [-I]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[1A]] + [[0A, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (14) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (15) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 0^1(1(0(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[-I], [-I], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[-I], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, 0A], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, 0A], [0A, -I, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (16) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (17) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(1(2(0(x1)))) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, -I, 0A], [-I, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[1A], [0A], [0A]] + [[0A, -I, 0A], [0A, 0A, -I], [0A, -I, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, 0A], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [1A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [1A], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [0A, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (18) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (19) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 5^1(5(0(3(1(x1))))) -> 5^1(0(x1)) 996.90/254.63 5^1(5(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [1A], [0A]] + [[0A, -I, 0A], [1A, 1A, 1A], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[0A]] + [[-I, -I, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [-I], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (20) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(3(1(x1)))) -> 5^1(x1) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 5^1(x1) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 5^1(0(x1)) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (21) DependencyGraphProof (EQUIVALENT) 996.90/254.63 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (22) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (23) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 1^1(0(3(1(x1)))) -> 0^1(1(1(x1))) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[-I], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[1A], [-I], [-I]] + [[0A, 0A, 0A], [0A, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (24) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (25) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 1^1(0(2(1(0(x1))))) -> 0^1(0(x1)) 996.90/254.63 0^1(1(0(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 0^1(x1) 996.90/254.63 0^1(5(4(1(3(x1))))) -> 0^1(x1) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [1A], [-I]] + [[0A, -I, -I], [-I, 0A, -I], [1A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, -I], [-I, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [-I], [1A]] + [[-I, -I, -I], [-I, 0A, 0A], [-I, -I, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 996.90/254.63 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (26) 996.90/254.63 Obligation: 996.90/254.63 Q DP problem: 996.90/254.63 The TRS P consists of the following rules: 996.90/254.63 996.90/254.63 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.63 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.63 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.63 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.63 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.63 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.63 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.63 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.63 996.90/254.63 The TRS R consists of the following rules: 996.90/254.63 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.63 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.63 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.63 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.63 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.63 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.63 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.63 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.63 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.63 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.63 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.63 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.63 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.63 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.63 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.63 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.63 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.63 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.63 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.63 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.63 996.90/254.63 Q is empty. 996.90/254.63 We have to consider all minimal (P,Q,R)-chains. 996.90/254.63 ---------------------------------------- 996.90/254.63 996.90/254.63 (27) QDPOrderProof (EQUIVALENT) 996.90/254.63 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.63 996.90/254.63 996.90/254.63 The following pairs can be oriented strictly and are deleted. 996.90/254.63 996.90/254.63 0^1(5(1(3(x1)))) -> 1^1(1(x1)) 996.90/254.63 The remaining pairs can at least be oriented weakly. 996.90/254.63 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(0(x_1)) = [[0A], [0A], [1A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(1^1(x_1)) = [[0A]] + [[0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(3(x_1)) = [[0A], [0A], [-I]] + [[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, -I], [-I, 0A, -I], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(5(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 <<< 996.90/254.63 POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 996.90/254.63 >>> 996.90/254.63 996.90/254.63 996.90/254.63 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.63 996.90/254.63 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.63 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.63 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.63 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.63 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (28) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.64 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.64 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.64 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.64 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (29) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 1^1(0(3(1(x1)))) -> 1^1(1(x1)) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0^1(x_1)) = [[1A]] + [[0A, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1(x_1)) = [[-I], [0A], [1A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0(x_1)) = [[0A], [-I], [1A]] + [[0A, -I, 0A], [0A, -I, 0A], [0A, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(3(x_1)) = [[-I], [-I], [0A]] + [[0A, -I, 0A], [-I, 0A, 0A], [-I, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(2(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [-I, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, 0A, 0A], [-I, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (30) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.64 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.64 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.64 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (31) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 1^1(0(2(1(0(x1))))) -> 1^1(0(0(x1))) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0^1(x_1)) = [[0A]] + [[0A, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1(x_1)) = [[0A], [1A], [0A]] + [[0A, 0A, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0(x_1)) = [[-I], [-I], [-I]] + [[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1^1(x_1)) = [[1A]] + [[0A, 0A, 1A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(2(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(3(x_1)) = [[-I], [0A], [0A]] + [[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(5(x_1)) = [[0A], [0A], [0A]] + [[-I, 0A, 0A], [-I, -I, 0A], [0A, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(4(x_1)) = [[0A], [0A], [0A]] + [[-I, -I, -I], [-I, -I, -I], [-I, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (32) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.64 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.64 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (33) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 1^1(2(0(3(1(x1))))) -> 0^1(x1) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1(x_1)) = [[1A], [1A], [0A]] + [[-I, 1A, -I], [-I, 1A, 0A], [-I, 0A, 1A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0(x_1)) = [[0A], [-I], [-I]] + [[-I, 0A, 0A], [0A, 0A, 0A], [-I, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1^1(x_1)) = [[1A]] + [[-I, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(2(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, 0A], [0A, -I, 0A], [-I, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(3(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(5(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(4(x_1)) = [[0A], [0A], [0A]] + [[0A, -I, -I], [0A, -I, -I], [0A, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (34) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(1(0(x1))) -> 1^1(x1) 996.90/254.64 0^1(1(0(x1))) -> 1^1(2(x1)) 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 0^1(1(0(3(x1)))) -> 1^1(0(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(1(3(x1)))) -> 1^1(x1) 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 1^1(x1) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (35) DependencyGraphProof (EQUIVALENT) 996.90/254.64 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (36) 996.90/254.64 Complex Obligation (AND) 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (37) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (38) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 1^1(2(0(3(1(x1))))) -> 1^1(2(0(x1))) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1^1(x_1)) = [[0A]] + [[-I, 0A, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(2(x_1)) = [[0A], [-I], [0A]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(0(x_1)) = [[0A], [-I], [-I]] + [[0A, -I, -I], [0A, -I, 0A], [0A, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(3(x_1)) = [[0A], [0A], [-I]] + [[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(1(x_1)) = [[1A], [1A], [-I]] + [[1A, 0A, 1A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(4(x_1)) = [[0A], [0A], [-I]] + [[-I, -I, -I], [-I, 0A, -I], [-I, -I, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 <<< 996.90/254.64 POL(5(x_1)) = [[0A], [0A], [-I]] + [[-I, 0A, 0A], [-I, 0A, 0A], [-I, 0A, 0A]] * x_1 996.90/254.64 >>> 996.90/254.64 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (39) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 P is empty. 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (40) PisEmptyProof (EQUIVALENT) 996.90/254.64 The TRS P is empty. Hence, there is no (P,Q,R) chain. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (41) 996.90/254.64 YES 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (42) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (43) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 0^1(0(1(3(x1)))) -> 0^1(1(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(1(x1)) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 996.90/254.64 996.90/254.64 POL( 0^1_1(x_1) ) = max{0, 2x_1 - 2} 996.90/254.64 POL( 5_1(x_1) ) = x_1 + 1 996.90/254.64 POL( 0_1(x_1) ) = x_1 + 1 996.90/254.64 POL( 3_1(x_1) ) = max{0, x_1 - 2} 996.90/254.64 POL( 1_1(x_1) ) = 2 996.90/254.64 POL( 4_1(x_1) ) = max{0, 2x_1 - 2} 996.90/254.64 POL( 2_1(x_1) ) = max{0, -2} 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (44) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (45) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 0^1(0(4(1(3(x1))))) -> 0^1(0(1(x1))) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 996.90/254.64 996.90/254.64 POL( 0^1_1(x_1) ) = 2x_1 996.90/254.64 POL( 5_1(x_1) ) = x_1 + 1 996.90/254.64 POL( 0_1(x_1) ) = x_1 + 1 996.90/254.64 POL( 3_1(x_1) ) = max{0, x_1 - 2} 996.90/254.64 POL( 1_1(x_1) ) = 2 996.90/254.64 POL( 4_1(x_1) ) = max{0, 2x_1 - 1} 996.90/254.64 POL( 2_1(x_1) ) = max{0, x_1 - 1} 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (46) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 The TRS P consists of the following rules: 996.90/254.64 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (47) QDPOrderProof (EQUIVALENT) 996.90/254.64 We use the reduction pair processor [LPAR04,JAR06]. 996.90/254.64 996.90/254.64 996.90/254.64 The following pairs can be oriented strictly and are deleted. 996.90/254.64 996.90/254.64 0^1(5(5(0(3(x1))))) -> 0^1(5(x1)) 996.90/254.64 The remaining pairs can at least be oriented weakly. 996.90/254.64 Used ordering: Polynomial interpretation [POLO]: 996.90/254.64 996.90/254.64 POL(0(x_1)) = x_1 996.90/254.64 POL(0^1(x_1)) = x_1 996.90/254.64 POL(1(x_1)) = 0 996.90/254.64 POL(2(x_1)) = 0 996.90/254.64 POL(3(x_1)) = x_1 996.90/254.64 POL(4(x_1)) = 0 996.90/254.64 POL(5(x_1)) = 1 + x_1 996.90/254.64 996.90/254.64 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 996.90/254.64 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 996.90/254.64 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (48) 996.90/254.64 Obligation: 996.90/254.64 Q DP problem: 996.90/254.64 P is empty. 996.90/254.64 The TRS R consists of the following rules: 996.90/254.64 996.90/254.64 0(1(0(x1))) -> 1(0(0(2(x1)))) 996.90/254.64 0(1(0(x1))) -> 0(2(2(1(0(x1))))) 996.90/254.64 0(1(0(x1))) -> 1(3(0(0(2(x1))))) 996.90/254.64 0(1(0(x1))) -> 3(0(4(2(1(0(x1)))))) 996.90/254.64 0(1(0(x1))) -> 4(0(0(2(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(3(2(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(0(2(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(2(0(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(4(4(3(1(x1)))))) 996.90/254.64 0(1(0(x1))) -> 2(2(1(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(2(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(3(0(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(4(0(2(0(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 1(0(4(0(4(2(x1)))))) 996.90/254.64 0(1(0(x1))) -> 0(0(2(2(1(4(x1)))))) 996.90/254.64 0(1(1(0(x1)))) -> 1(1(0(0(2(2(x1)))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(0(x1))))) 996.90/254.64 0(5(1(0(x1)))) -> 0(1(5(3(3(0(x1)))))) 996.90/254.64 0(5(5(0(x1)))) -> 5(5(0(0(2(2(x1)))))) 996.90/254.64 0(1(0(1(x1)))) -> 0(0(5(3(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 1(4(3(0(1(1(x1)))))) 996.90/254.64 1(0(3(1(x1)))) -> 0(4(3(1(1(5(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 4(0(2(3(1(0(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(3(4(0(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(0(3(2(2(x1)))))) 996.90/254.64 0(1(0(3(x1)))) -> 1(0(2(2(0(3(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(4(3(0(1(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(1(3(0(2(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 1(5(0(0(3(x1))))) 996.90/254.64 0(0(1(3(x1)))) -> 2(0(3(0(2(1(x1)))))) 996.90/254.64 0(0(1(3(x1)))) -> 0(2(0(3(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(1(1(x1))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(2(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(4(3(1(3(0(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 0(5(3(2(5(1(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(2(0(4(1(3(x1)))))) 996.90/254.64 0(5(1(3(x1)))) -> 5(3(1(5(0(4(x1)))))) 996.90/254.64 0(1(2(0(0(x1))))) -> 1(3(0(0(0(2(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(2(3(1(0(0(x1)))))) 996.90/254.64 1(0(2(1(0(x1))))) -> 1(1(3(0(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 2(3(1(1(2(0(x1)))))) 996.90/254.64 1(2(0(3(1(x1))))) -> 1(1(2(2(3(0(x1)))))) 996.90/254.64 5(5(0(3(1(x1))))) -> 1(5(3(4(5(0(x1)))))) 996.90/254.64 0(5(1(3(1(x1))))) -> 0(2(5(1(3(1(x1)))))) 996.90/254.64 0(0(5(0(3(x1))))) -> 0(5(3(0(0(3(x1)))))) 996.90/254.64 0(5(5(0(3(x1))))) -> 0(5(3(2(0(5(x1)))))) 996.90/254.64 0(0(4(1(3(x1))))) -> 0(4(3(0(0(1(x1)))))) 996.90/254.64 0(5(4(1(3(x1))))) -> 4(5(3(1(5(0(x1)))))) 996.90/254.64 996.90/254.64 Q is empty. 996.90/254.64 We have to consider all minimal (P,Q,R)-chains. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (49) PisEmptyProof (EQUIVALENT) 996.90/254.64 The TRS P is empty. Hence, there is no (P,Q,R) chain. 996.90/254.64 ---------------------------------------- 996.90/254.64 996.90/254.64 (50) 996.90/254.64 YES 997.04/254.71 EOF