YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 181 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 1200 ms] (4) QDP (5) QDPOrderProof [EQUIVALENT, 1713 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 0 ms] (8) AND (9) QDP (10) QDPOrderProof [EQUIVALENT, 69 ms] (11) QDP (12) DependencyGraphProof [EQUIVALENT, 0 ms] (13) QDP (14) QDPOrderProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 2682 ms] (17) QDP (18) PisEmptyProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) QDPOrderProof [EQUIVALENT, 57 ms] (22) QDP (23) QDPOrderProof [EQUIVALENT, 0 ms] (24) QDP (25) PisEmptyProof [EQUIVALENT, 0 ms] (26) YES (27) QDP (28) QDPOrderProof [EQUIVALENT, 65 ms] (29) QDP (30) QDPOrderProof [EQUIVALENT, 0 ms] (31) QDP (32) QDPOrderProof [EQUIVALENT, 49 ms] (33) QDP (34) PisEmptyProof [EQUIVALENT, 0 ms] (35) YES (36) QDP (37) QDPOrderProof [EQUIVALENT, 53 ms] (38) QDP (39) QDPOrderProof [EQUIVALENT, 60 ms] (40) QDP (41) QDPOrderProof [EQUIVALENT, 50 ms] (42) QDP (43) PisEmptyProof [EQUIVALENT, 0 ms] (44) YES (45) QDP (46) QDPOrderProof [EQUIVALENT, 56 ms] (47) QDP (48) QDPOrderProof [EQUIVALENT, 0 ms] (49) QDP (50) QDPOrderProof [EQUIVALENT, 60 ms] (51) QDP (52) QDPOrderProof [EQUIVALENT, 1720 ms] (53) QDP (54) PisEmptyProof [EQUIVALENT, 0 ms] (55) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 3(2(3(1(2(2(2(4(3(5(x1)))))))))) -> 5(3(4(4(4(4(0(4(0(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 4(2(3(1(4(3(0(3(1(1(5(0(4(1(3(4(3(1(4(4(x1)))))))))))))))))))) -> 4(1(4(4(2(5(5(1(5(4(5(5(2(0(5(0(2(0(5(x1))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(4(x_1)) = 1 + x_1 POL(5(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3(2(3(1(2(2(2(4(3(5(x1)))))))))) -> 5(3(4(4(4(4(0(4(0(x1))))))))) 4(2(3(1(4(3(0(3(1(1(5(0(4(1(3(4(3(1(4(4(x1)))))))))))))))))))) -> 4(1(4(4(2(5(5(1(5(4(5(5(2(0(5(0(2(0(5(x1))))))))))))))))))) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1)))) 0^1(0(1(2(x1)))) -> 3^1(4(2(x1))) 0^1(0(1(2(x1)))) -> 4^1(2(x1)) 2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1)))))) 2^1(0(1(0(5(3(x1)))))) -> 0^1(3(1(3(3(x1))))) 2^1(0(1(0(5(3(x1)))))) -> 3^1(1(3(3(x1)))) 2^1(0(1(0(5(3(x1)))))) -> 1^1(3(3(x1))) 2^1(0(1(0(5(3(x1)))))) -> 3^1(3(x1)) 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) 4^1(1(1(5(0(4(x1)))))) -> 4^1(2(2(0(4(x1))))) 4^1(1(1(5(0(4(x1)))))) -> 2^1(2(0(4(x1)))) 4^1(1(1(5(0(4(x1)))))) -> 2^1(0(4(x1))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 0^1(2(1(4(2(5(3(2(x1)))))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(1(4(2(5(3(2(x1))))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 1^1(4(2(5(3(2(x1)))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 4^1(2(5(3(2(x1))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(5(3(2(x1)))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 5^1(3(2(x1))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 3^1(2(x1)) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(x1) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1)))))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(1(3(0(3(0(3(x1))))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 1^1(3(0(3(0(3(x1)))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(0(3(x1))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(0(3(x1)))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(x1))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(x1)) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(3(1(0(4(1(0(3(3(x1))))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(1(0(4(1(0(3(3(x1)))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(4(1(0(3(3(x1))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(4(1(0(3(3(x1)))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(1(0(3(3(x1))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(3(3(x1)))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(3(3(x1))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(3(x1)) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(x1) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1))))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(5(3(1(2(3(4(2(x1)))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 5^1(3(1(2(3(4(2(x1))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(1(2(3(4(2(x1)))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 1^1(2(3(4(2(x1))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 2^1(3(4(2(x1)))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(4(2(x1))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(4(1(1(3(5(3(4(2(x1)))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(4(1(1(3(5(3(4(2(x1))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(1(1(3(5(3(4(2(x1)))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(1(3(5(3(4(2(x1))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(3(5(3(4(2(x1)))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(5(3(4(2(x1))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 5^1(3(4(2(x1)))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(4(2(x1))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(2(x1)) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 2^1(x1) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(2(3(3(3(3(0(1(1(3(4(x1))))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(3(3(3(3(0(1(1(3(4(x1)))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(3(0(1(1(3(4(x1))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(0(1(1(3(4(x1)))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(0(1(1(3(4(x1))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(0(1(1(3(4(x1)))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 0^1(1(1(3(4(x1))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(1(3(4(x1)))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(3(4(x1))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(4(x1)) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 4^1(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(2(3(3(2(1(2(5(0(5(x1))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(2(3(3(2(1(2(5(0(5(x1)))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(3(3(2(1(2(5(0(5(x1))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(3(2(1(2(5(0(5(x1)))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(1(2(5(0(5(x1))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(1(2(5(0(5(x1)))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 1^1(2(5(0(5(x1))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(5(0(5(x1)))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(0(5(x1))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 0^1(5(x1)) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(x1) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(4(0(3(2(4(1(0(2(4(x1))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(0(3(2(4(1(0(2(4(x1)))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(3(2(4(1(0(2(4(x1))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(3(2(4(1(0(2(4(x1)))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 3^1(2(4(1(0(2(4(x1))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(1(0(2(4(x1)))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(0(2(4(x1))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(0(2(4(x1)))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(2(4(x1))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(x1)) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(0(1(5(1(1(4(1(0(3(4(x1))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(1(5(1(1(4(1(0(3(4(x1)))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(5(1(1(4(1(0(3(4(x1))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(1(1(4(1(0(3(4(x1)))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(1(4(1(0(3(4(x1))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(4(1(0(3(4(x1)))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(1(0(3(4(x1))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(0(3(4(x1)))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(3(4(x1))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 3^1(4(x1)) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(x1) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(5(0(3(1(0(4(3(1(3(2(x1))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(0(3(1(0(4(3(1(3(2(x1)))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(3(1(0(4(3(1(3(2(x1))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(0(4(3(1(3(2(x1)))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(0(4(3(1(3(2(x1))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(4(3(1(3(2(x1)))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 4^1(3(1(3(2(x1))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(3(2(x1)))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(3(2(x1))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(3(5(4(5(5(0(0(3(1(x1))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(3(5(4(5(5(0(0(3(1(x1)))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(5(5(0(0(3(1(x1))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(5(5(0(0(3(1(x1)))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(5(5(0(0(3(1(x1))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(5(0(0(3(1(x1)))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(0(0(3(1(x1))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(0(3(1(x1)))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(3(1(x1))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(1(x1)) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 1^1(x1) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 2^1(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(0(5(4(1(3(5(4(4(1(0(x1))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(5(4(1(3(5(4(4(1(0(x1)))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(1(3(5(4(4(1(0(x1))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(3(5(4(4(1(0(x1)))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(3(5(4(4(1(0(x1))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(5(4(4(1(0(x1)))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(4(1(0(x1))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(1(0(x1)))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(0(x1))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(0(x1)) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(x1) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(2(1(4(3(0(2(2(5(5(x1))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(1(4(3(0(2(2(5(5(x1)))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(4(3(0(2(2(5(5(x1))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(4(3(0(2(2(5(5(x1)))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(0(2(2(5(5(x1))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(0(2(2(5(5(x1)))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 0^1(2(2(5(5(x1))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(5(5(x1)))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(5(5(x1))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(5(x1)) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(x1) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(5(0(3(1(2(1(1(2(4(3(x1))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(3(1(2(1(1(2(4(3(x1)))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(3(1(2(1(1(2(4(3(x1))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(2(1(1(2(4(3(x1)))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(1(1(2(4(3(x1))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(1(1(2(4(3(x1)))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(1(2(4(3(x1))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(4(3(x1)))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(4(3(x1))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(3(x1)) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(x1) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(2(2(2(1(3(2(1(2(2(2(x1))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(1(3(2(1(2(2(2(x1)))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(1(3(2(1(2(2(2(x1))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(3(2(1(2(2(2(x1)))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(3(2(1(2(2(2(x1))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 3^1(2(1(2(2(2(x1)))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(2(2(2(x1))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(2(2(2(x1)))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(x1))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(x1)) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(1(5(2(0(5(4(4(2(1(4(x1))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(5(2(0(5(4(4(2(1(4(x1)))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(2(0(5(4(4(2(1(4(x1))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(0(5(4(4(2(1(4(x1)))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(5(4(4(2(1(4(x1))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(4(4(2(1(4(x1)))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(4(2(1(4(x1))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(1(4(x1)))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(4(x1))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(4(x1)) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(5(0(2(3(0(5(2(1(5(1(x1))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(0(2(3(0(5(2(1(5(1(x1)))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(3(0(5(2(1(5(1(x1))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(3(0(5(2(1(5(1(x1)))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(0(5(2(1(5(1(x1))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(5(2(1(5(1(x1)))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(2(1(5(1(x1))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(1(5(1(x1)))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(5(1(x1))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(4(2(0(3(5(1(0(3(2(x1))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(4(2(0(3(5(1(0(3(2(x1)))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 4^1(2(0(3(5(1(0(3(2(x1))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(0(3(5(1(0(3(2(x1)))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(1(0(3(2(x1))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(1(0(3(2(x1)))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(0(3(2(x1))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(0(3(2(x1)))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(2(x1))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(x1)) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(0(5(1(4(2(3(2(3(0(0(x1))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(5(1(4(2(3(2(3(0(0(x1)))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 5^1(1(4(2(3(2(3(0(0(x1))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 1^1(4(2(3(2(3(0(0(x1)))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(3(2(3(0(0(x1))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(2(3(0(0(x1)))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(2(3(0(0(x1))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(0(0(x1)))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(0(x1))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(0(x1)) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 3^1(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(0(5(0(1(2(5(2(4(5(0(x1))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(5(0(1(2(5(2(4(5(0(x1)))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(1(2(5(2(4(5(0(x1))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(1(2(5(2(4(5(0(x1)))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(2(5(2(4(5(0(x1))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(5(2(4(5(0(x1)))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(5(0(x1))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(0(x1)))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(0(x1))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(x1)) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(x1) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(1(2(x1)))) -> 4^1(2(x1)) 2^1(0(1(0(5(3(x1)))))) -> 0^1(3(1(3(3(x1))))) 2^1(0(1(0(5(3(x1)))))) -> 3^1(1(3(3(x1)))) 2^1(0(1(0(5(3(x1)))))) -> 1^1(3(3(x1))) 2^1(0(1(0(5(3(x1)))))) -> 3^1(3(x1)) 4^1(1(1(5(0(4(x1)))))) -> 4^1(2(2(0(4(x1))))) 4^1(1(1(5(0(4(x1)))))) -> 2^1(2(0(4(x1)))) 4^1(1(1(5(0(4(x1)))))) -> 2^1(0(4(x1))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 0^1(2(1(4(2(5(3(2(x1)))))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(1(4(2(5(3(2(x1))))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 1^1(4(2(5(3(2(x1)))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 4^1(2(5(3(2(x1))))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(5(3(2(x1)))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 5^1(3(2(x1))) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 3^1(2(x1)) 2^1(3(2(5(5(0(0(0(x1)))))))) -> 2^1(x1) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 1^1(3(0(3(0(3(x1)))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(0(3(x1))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(0(3(x1)))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(0(3(x1))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 0^1(3(x1)) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(1(0(4(1(0(3(3(x1)))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(4(1(0(3(3(x1))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(4(1(0(3(3(x1)))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(1(0(3(3(x1))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 1^1(0(3(3(x1)))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 0^1(3(3(x1))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(3(x1)) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 3^1(x1) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(5(3(1(2(3(4(2(x1)))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 5^1(3(1(2(3(4(2(x1))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(1(2(3(4(2(x1)))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 1^1(2(3(4(2(x1))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 2^1(3(4(2(x1)))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 3^1(4(2(x1))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(4(1(1(3(5(3(4(2(x1)))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(4(1(1(3(5(3(4(2(x1))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(1(1(3(5(3(4(2(x1)))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(1(3(5(3(4(2(x1))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 1^1(3(5(3(4(2(x1)))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(5(3(4(2(x1))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 5^1(3(4(2(x1)))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 3^1(4(2(x1))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 4^1(2(x1)) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 2^1(x1) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(2(3(3(3(3(0(1(1(3(4(x1))))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 2^1(3(3(3(3(0(1(1(3(4(x1)))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(3(0(1(1(3(4(x1))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(3(0(1(1(3(4(x1)))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(3(0(1(1(3(4(x1))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(0(1(1(3(4(x1)))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 0^1(1(1(3(4(x1))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(1(3(4(x1)))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(3(4(x1))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 3^1(4(x1)) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 4^1(3(2(2(3(3(2(1(2(5(0(5(x1)))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(2(3(3(2(1(2(5(0(5(x1))))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(2(3(3(2(1(2(5(0(5(x1)))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(3(3(2(1(2(5(0(5(x1))))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(3(2(1(2(5(0(5(x1)))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 3^1(2(1(2(5(0(5(x1))))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(1(2(5(0(5(x1)))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 1^1(2(5(0(5(x1))))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 2^1(5(0(5(x1)))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(0(5(x1))) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 0^1(5(x1)) 4^1(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5^1(x1) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(4(0(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(4(1(4(0(3(2(4(1(0(2(4(x1)))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(4(0(3(2(4(1(0(2(4(x1))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(0(3(2(4(1(0(2(4(x1)))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(0(3(2(4(1(0(2(4(x1))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(3(2(4(1(0(2(4(x1)))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 3^1(2(4(1(0(2(4(x1))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(1(0(2(4(x1)))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 4^1(1(0(2(4(x1))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(0(2(4(x1)))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 0^1(2(4(x1))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 2^1(4(x1)) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(5(4(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(2(0(1(5(1(1(4(1(0(3(4(x1)))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(0(1(5(1(1(4(1(0(3(4(x1))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(1(5(1(1(4(1(0(3(4(x1)))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(5(1(1(4(1(0(3(4(x1))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 5^1(1(1(4(1(0(3(4(x1)))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(1(4(1(0(3(4(x1))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(4(1(0(3(4(x1)))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(1(0(3(4(x1))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 1^1(0(3(4(x1)))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 0^1(3(4(x1))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 3^1(4(x1)) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 4^1(x1) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(3(3(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(5(0(3(1(0(4(3(1(3(2(x1)))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(5(0(3(1(0(4(3(1(3(2(x1))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(0(3(1(0(4(3(1(3(2(x1)))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(3(1(0(4(3(1(3(2(x1))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(0(4(3(1(3(2(x1)))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(0(4(3(1(3(2(x1))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 0^1(4(3(1(3(2(x1)))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 4^1(3(1(3(2(x1))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 3^1(1(3(2(x1)))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 1^1(3(2(x1))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(3(5(4(3(5(4(5(5(0(0(3(1(x1))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(3(5(4(5(5(0(0(3(1(x1))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(3(5(4(5(5(0(0(3(1(x1)))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(5(4(5(5(0(0(3(1(x1))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(4(5(5(0(0(3(1(x1)))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 4^1(5(5(0(0(3(1(x1))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(5(0(0(3(1(x1)))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 5^1(0(0(3(1(x1))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(0(3(1(x1)))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 0^1(3(1(x1))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 3^1(1(x1)) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 1^1(x1) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 2^1(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(0(5(4(1(3(5(4(4(1(0(x1))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(5(4(1(3(5(4(4(1(0(x1)))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(1(3(5(4(4(1(0(x1))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(3(5(4(4(1(0(x1)))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(3(5(4(4(1(0(x1))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 3^1(5(4(4(1(0(x1)))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 5^1(4(4(1(0(x1))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(1(0(x1)))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(1(0(x1))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 1^1(0(x1)) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(x1) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(3(2(2(1(4(3(0(2(2(5(5(x1)))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(2(2(1(4(3(0(2(2(5(5(x1))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(1(4(3(0(2(2(5(5(x1)))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(1(4(3(0(2(2(5(5(x1))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 1^1(4(3(0(2(2(5(5(x1)))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 4^1(3(0(2(2(5(5(x1))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 3^1(0(2(2(5(5(x1)))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 0^1(2(2(5(5(x1))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(5(5(x1)))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(5(5(x1))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(5(x1)) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 5^1(x1) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(4(5(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(5(0(3(1(2(1(1(2(4(3(x1)))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(5(0(3(1(2(1(1(2(4(3(x1))))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5^1(0(3(1(2(1(1(2(4(3(x1)))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 0^1(3(1(2(1(1(2(4(3(x1))))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(1(2(1(1(2(4(3(x1)))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(1(1(2(4(3(x1))))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(1(1(2(4(3(x1)))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(1(2(4(3(x1))))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 1^1(2(4(3(x1)))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 2^1(4(3(x1))) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 4^1(3(x1)) 2^1(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 3^1(x1) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(1(0(2(2(2(1(3(2(1(2(2(2(x1))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 0^1(2(2(2(1(3(2(1(2(2(2(x1))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(1(3(2(1(2(2(2(x1)))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(1(3(2(1(2(2(2(x1))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(3(2(1(2(2(2(x1)))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(3(2(1(2(2(2(x1))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 3^1(2(1(2(2(2(x1)))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(1(2(2(2(x1))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 1^1(2(2(2(x1)))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(2(x1))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 2^1(2(x1)) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(2(1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(0(1(5(2(0(5(4(4(2(1(4(x1)))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(1(5(2(0(5(4(4(2(1(4(x1))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(5(2(0(5(4(4(2(1(4(x1)))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(2(0(5(4(4(2(1(4(x1))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(0(5(4(4(2(1(4(x1)))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 0^1(5(4(4(2(1(4(x1))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 5^1(4(4(2(1(4(x1)))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(4(2(1(4(x1))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 4^1(2(1(4(x1)))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 2^1(1(4(x1))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 1^1(4(x1)) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 4^1(5(0(2(3(0(5(2(1(5(1(x1))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(0(2(3(0(5(2(1(5(1(x1)))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(2(3(0(5(2(1(5(1(x1))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(3(0(5(2(1(5(1(x1)))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(0(5(2(1(5(1(x1))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(5(2(1(5(1(x1)))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 5^1(2(1(5(1(x1))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 2^1(1(5(1(x1)))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 1^1(5(1(x1))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(5(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(3(2(4(2(0(3(5(1(0(3(2(x1)))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(4(2(0(3(5(1(0(3(2(x1))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(4(2(0(3(5(1(0(3(2(x1)))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 4^1(2(0(3(5(1(0(3(2(x1))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 2^1(0(3(5(1(0(3(2(x1)))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(5(1(0(3(2(x1))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(5(1(0(3(2(x1)))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 5^1(1(0(3(2(x1))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(0(3(2(x1)))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 0^1(3(2(x1))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 3^1(2(x1)) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(4(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(0(5(1(4(2(3(2(3(0(0(x1)))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(0(5(1(4(2(3(2(3(0(0(x1))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(5(1(4(2(3(2(3(0(0(x1)))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 5^1(1(4(2(3(2(3(0(0(x1))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 1^1(4(2(3(2(3(0(0(x1)))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(2(3(2(3(0(0(x1))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(2(3(0(0(x1)))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(2(3(0(0(x1))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 2^1(3(0(0(x1)))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 3^1(0(0(x1))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 0^1(0(x1)) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 3^1(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(4(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(0(5(0(1(2(5(2(4(5(0(x1)))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(0(5(0(1(2(5(2(4(5(0(x1))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(5(0(1(2(5(2(4(5(0(x1)))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(1(2(5(2(4(5(0(x1))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(1(2(5(2(4(5(0(x1)))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 1^1(2(5(2(4(5(0(x1))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(5(2(4(5(0(x1)))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(2(4(5(0(x1))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 2^1(4(5(0(x1)))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(5(0(x1))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 5^1(0(x1)) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 0^1(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 + x_1 POL(1^1(x_1)) = 1 + x_1 POL(2(x_1)) = 1 + x_1 POL(2^1(x_1)) = 1 + x_1 POL(3(x_1)) = 1 + x_1 POL(3^1(x_1)) = 1 + x_1 POL(4(x_1)) = 1 + x_1 POL(4^1(x_1)) = 1 + x_1 POL(5(x_1)) = 1 + x_1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1)))) 0^1(0(1(2(x1)))) -> 3^1(4(2(x1))) 2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1)))))) 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1)))))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 3^1(1(3(0(3(0(3(x1))))))) 1^1(0(0(1(5(5(4(2(4(x1))))))))) -> 4^1(3(1(0(4(1(0(3(3(x1))))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1))))))))) 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 4^1(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1)))))))))))))))) 1^1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2^1(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 3^1(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1))))))))))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2^1(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4^1(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 7 less nodes. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1))))))))) 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3^1(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = x_1 POL(2(x_1)) = 1 POL(3(x_1)) = 0 POL(3^1(x_1)) = 0 POL(4(x_1)) = x_1 POL(4^1(x_1)) = x_1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: 3^1(0(0(1(0(4(5(4(2(x1))))))))) -> 4^1(4(5(3(1(2(3(4(2(x1))))))))) 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(1(1(5(0(4(x1)))))) -> 4^1(4(2(2(0(4(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(4^1(x_1)) = x_1 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 4^1(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4^1(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 4^1_1(x_1) ) = 2x_1 + 1 POL( 0_1(x_1) ) = 2 POL( 1_1(x_1) ) = max{0, -2} POL( 2_1(x_1) ) = x_1 POL( 5_1(x_1) ) = max{0, 2x_1 - 2} POL( 4_1(x_1) ) = max{0, x_1 - 1} POL( 3_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) ---------------------------------------- (17) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1)))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(0(0(5(3(3(4(3(x1)))))))) -> 5^1(3(1(3(0(3(0(3(x1)))))))) 5^1(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5^1(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 5^1(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5^1(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 0 POL(2(x_1)) = 1 + x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 1 POL(5^1(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (24) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (26) YES ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1)))))) 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2^1(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1)))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(0(1(0(5(3(x1)))))) -> 2^1(0(3(1(3(3(x1)))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2^1(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(2^1(x_1)) = x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) ---------------------------------------- (33) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (35) YES ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1^1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1^1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(1(x_1)) = x_1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = 1 POL(3(x_1)) = 0 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1^1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(1^1(x_1)) = x_1 POL(2(x_1)) = x_1 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) ---------------------------------------- (42) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1)))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(1(2(x1)))) -> 0^1(3(4(2(x1)))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 + x_1 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0^1(0(0(4(1(1(3(5(3(4(2(x1))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = 0 POL(3(x_1)) = 0 POL(4(x_1)) = 0 POL(5(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (50) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0^1(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0(x_1)) = 1 POL(0^1(x_1)) = x_1 POL(1(x_1)) = 1 POL(2(x_1)) = 1 POL(3(x_1)) = 0 POL(4(x_1)) = x_1 POL(5(x_1)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0^1(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = max{0, 2x_1 - 2} POL( 3_1(x_1) ) = max{0, x_1 - 2} POL( 0_1(x_1) ) = 2 POL( 4_1(x_1) ) = max{0, -2} POL( 5_1(x_1) ) = max{0, 2x_1 - 2} POL( 2_1(x_1) ) = 2x_1 POL( 1_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) ---------------------------------------- (53) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(0(1(2(x1)))) -> 0(3(4(2(x1)))) 2(0(1(0(5(3(x1)))))) -> 2(0(3(1(3(3(x1)))))) 4(1(1(5(0(4(x1)))))) -> 4(4(2(2(0(4(x1)))))) 2(3(2(5(5(0(0(0(x1)))))))) -> 0(2(1(4(2(5(3(2(x1)))))))) 5(0(0(5(3(3(4(3(x1)))))))) -> 5(3(1(3(0(3(0(3(x1)))))))) 1(0(0(1(5(5(4(2(4(x1))))))))) -> 4(3(1(0(4(1(0(3(3(x1))))))))) 3(0(0(1(0(4(5(4(2(x1))))))))) -> 4(4(5(3(1(2(3(4(2(x1))))))))) 0(0(1(2(5(1(4(4(5(5(4(x1))))))))))) -> 0(0(0(4(1(1(3(5(3(4(2(x1))))))))))) 1(5(2(2(5(1(1(2(4(4(5(4(x1)))))))))))) -> 1(2(2(3(3(3(3(0(1(1(3(4(x1)))))))))))) 4(1(5(3(3(5(0(4(5(1(4(4(1(x1))))))))))))) -> 5(4(3(2(2(3(3(2(1(2(5(0(5(x1))))))))))))) 1(0(3(0(0(0(0(2(3(1(4(0(2(0(4(x1))))))))))))))) -> 1(4(4(0(4(1(4(0(3(2(4(1(0(2(4(x1))))))))))))))) 2(5(0(5(1(5(2(3(5(3(1(3(5(4(5(x1))))))))))))))) -> 2(2(5(4(2(0(1(5(1(1(4(1(0(3(4(x1))))))))))))))) 5(2(5(1(1(5(0(0(3(0(5(5(3(3(2(x1))))))))))))))) -> 5(5(3(3(1(5(0(3(1(0(4(3(1(3(2(x1))))))))))))))) 2(1(1(5(2(5(5(0(1(3(4(1(4(4(5(3(x1)))))))))))))))) -> 2(5(2(3(3(5(4(3(5(4(5(5(0(0(3(1(x1)))))))))))))))) 0(4(4(5(0(4(5(0(5(2(3(0(0(2(4(4(2(x1))))))))))))))))) -> 0(4(4(3(0(2(5(0(5(4(1(3(5(4(4(1(0(x1))))))))))))))))) 1(1(0(0(3(1(3(0(2(1(2(5(2(5(0(2(0(x1))))))))))))))))) -> 2(2(4(3(2(1(3(2(2(1(4(3(0(2(2(5(5(x1))))))))))))))))) 2(5(4(4(4(2(3(5(4(0(4(0(0(5(4(5(5(x1))))))))))))))))) -> 5(3(1(4(4(5(0(5(0(3(1(2(1(1(2(4(3(x1))))))))))))))))) 5(0(0(1(3(3(2(5(3(4(0(3(4(3(5(4(4(2(x1)))))))))))))))))) -> 5(5(0(0(1(1(1(0(2(2(2(1(3(2(1(2(2(2(x1)))))))))))))))))) 4(4(0(2(2(1(1(4(1(1(1(0(0(0(2(3(4(4(4(x1))))))))))))))))))) -> 3(1(0(3(4(2(2(1(0(1(5(2(0(5(4(4(2(1(4(x1))))))))))))))))))) 0(3(2(2(1(4(5(3(3(5(5(1(5(1(0(5(1(5(5(1(x1)))))))))))))))))))) -> 0(3(1(4(0(2(0(0(2(4(5(0(2(3(0(5(2(1(5(1(x1)))))))))))))))))))) 1(4(1(4(2(5(4(5(5(0(5(0(4(0(4(5(0(0(5(4(2(x1))))))))))))))))))))) -> 1(5(5(1(1(3(0(3(5(5(3(2(4(2(0(3(5(1(0(3(2(x1))))))))))))))))))))) 2(2(4(0(0(1(4(3(1(4(4(3(1(2(1(1(5(0(4(5(0(x1))))))))))))))))))))) -> 4(3(0(2(4(0(4(3(0(4(2(0(5(1(4(2(3(2(3(0(0(x1))))))))))))))))))))) 4(2(4(2(5(5(2(0(2(0(2(0(3(1(5(3(0(1(4(4(2(x1))))))))))))))))))))) -> 4(2(4(5(1(3(5(2(4(4(2(0(5(0(1(2(5(2(4(5(0(x1))))))))))))))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (55) YES