/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 643 ms] (4) QDP (5) SplitQDPProof [EQUIVALENT, 0 ms] (6) AND (7) QDP (8) SemLabProof [SOUND, 0 ms] (9) QDP (10) DependencyGraphProof [EQUIVALENT, 4 ms] (11) AND (12) QDP (13) QDPOrderProof [EQUIVALENT, 247 ms] (14) QDP (15) QDPOrderProof [EQUIVALENT, 243 ms] (16) QDP (17) PisEmptyProof [SOUND, 0 ms] (18) TRUE (19) QDP (20) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (21) QDP (22) PisEmptyProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (26) QDP (27) PisEmptyProof [EQUIVALENT, 0 ms] (28) YES (29) QDP (30) QDPOrderProof [EQUIVALENT, 2837 ms] (31) QDP (32) DependencyGraphProof [EQUIVALENT, 0 ms] (33) QDP (34) QDPOrderProof [EQUIVALENT, 1758 ms] (35) QDP (36) PisEmptyProof [EQUIVALENT, 0 ms] (37) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 1(2(0(x1))) -> 4(0(3(3(5(4(5(1(4(3(x1)))))))))) 1(0(0(4(5(x1))))) -> 1(4(3(1(3(1(4(5(2(3(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(3(1(2(2(4(5(0(4(3(x1)))))))))) 2(1(0(1(0(x1))))) -> 3(5(4(5(4(3(3(1(1(2(x1)))))))))) 3(4(2(0(2(x1))))) -> 3(5(3(0(3(3(2(5(3(2(x1)))))))))) 0(3(5(2(4(0(x1)))))) -> 4(4(0(2(3(2(2(5(3(2(x1)))))))))) 1(1(2(0(4(5(x1)))))) -> 3(0(5(4(2(1(0(2(3(3(x1)))))))))) 2(1(1(0(1(2(x1)))))) -> 3(4(4(1(3(2(2(2(5(5(x1)))))))))) 2(2(0(1(1(1(x1)))))) -> 2(3(4(1(5(2(2(2(5(4(x1)))))))))) 2(4(1(0(4(2(x1)))))) -> 1(5(1(3(2(3(4(4(4(0(x1)))))))))) 2(4(2(1(1(1(x1)))))) -> 1(3(5(4(3(4(3(1(4(4(x1)))))))))) 3(0(1(0(0(2(x1)))))) -> 2(4(2(5(3(5(0(3(3(2(x1)))))))))) 3(0(1(1(1(1(x1)))))) -> 3(2(2(4(4(5(2(4(5(1(x1)))))))))) 4(1(1(2(0(2(x1)))))) -> 4(0(3(4(4(4(2(3(2(3(x1)))))))))) 0(2(1(1(1(1(0(x1))))))) -> 0(1(5(5(3(5(2(5(5(5(x1)))))))))) 0(2(4(1(1(1(5(x1))))))) -> 4(4(3(4(3(2(3(0(2(2(x1)))))))))) 0(4(2(0(0(4(1(x1))))))) -> 4(2(5(4(1(0(4(3(3(1(x1)))))))))) 0(4(3(0(5(4(1(x1))))))) -> 0(3(1(5(3(1(2(5(4(1(x1)))))))))) 1(0(5(2(2(0(0(x1))))))) -> 1(5(4(4(3(4(5(4(5(2(x1)))))))))) 1(1(3(4(5(0(0(x1))))))) -> 1(3(1(5(3(4(1(4(5(3(x1)))))))))) 1(4(3(1(5(0(5(x1))))))) -> 5(0(3(3(2(4(1(3(3(2(x1)))))))))) 1(5(0(2(0(5(5(x1))))))) -> 2(5(2(5(4(2(0(0(5(5(x1)))))))))) 2(0(1(5(2(0(5(x1))))))) -> 4(3(3(5(5(3(1(3(5(5(x1)))))))))) 2(4(0(5(4(1(4(x1))))))) -> 3(4(5(5(1(5(3(5(1(4(x1)))))))))) 3(4(1(4(0(4(5(x1))))))) -> 3(2(2(1(3(4(3(3(0(3(x1)))))))))) 4(1(0(4(2(0(0(x1))))))) -> 4(2(2(3(1(0(0(3(4(0(x1)))))))))) 4(1(0(4(2(0(3(x1))))))) -> 0(4(3(0(0(1(5(4(3(2(x1)))))))))) 4(1(1(1(0(1(2(x1))))))) -> 3(3(2(3(3(0(1(5(5(2(x1)))))))))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(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(2(1(x1))) -> 3^1(4(1(5(4(5(3(3(0(4(x1)))))))))) 0^1(2(1(x1))) -> 4^1(1(5(4(5(3(3(0(4(x1))))))))) 0^1(2(1(x1))) -> 1^1(5(4(5(3(3(0(4(x1)))))))) 0^1(2(1(x1))) -> 5^1(4(5(3(3(0(4(x1))))))) 0^1(2(1(x1))) -> 4^1(5(3(3(0(4(x1)))))) 0^1(2(1(x1))) -> 5^1(3(3(0(4(x1))))) 0^1(2(1(x1))) -> 3^1(3(0(4(x1)))) 0^1(2(1(x1))) -> 3^1(0(4(x1))) 0^1(2(1(x1))) -> 0^1(4(x1)) 0^1(2(1(x1))) -> 4^1(x1) 5^1(4(0(0(1(x1))))) -> 3^1(2(5(4(1(3(1(3(4(1(x1)))))))))) 5^1(4(0(0(1(x1))))) -> 2^1(5(4(1(3(1(3(4(1(x1))))))))) 5^1(4(0(0(1(x1))))) -> 5^1(4(1(3(1(3(4(1(x1)))))))) 5^1(4(0(0(1(x1))))) -> 4^1(1(3(1(3(4(1(x1))))))) 5^1(4(0(0(1(x1))))) -> 1^1(3(1(3(4(1(x1)))))) 5^1(4(0(0(1(x1))))) -> 3^1(1(3(4(1(x1))))) 5^1(4(0(0(1(x1))))) -> 1^1(3(4(1(x1)))) 5^1(4(0(0(1(x1))))) -> 3^1(4(1(x1))) 5^1(4(0(0(1(x1))))) -> 4^1(1(x1)) 2^1(0(3(0(2(x1))))) -> 3^1(4(0(5(4(2(2(1(3(3(x1)))))))))) 2^1(0(3(0(2(x1))))) -> 4^1(0(5(4(2(2(1(3(3(x1))))))))) 2^1(0(3(0(2(x1))))) -> 0^1(5(4(2(2(1(3(3(x1)))))))) 2^1(0(3(0(2(x1))))) -> 5^1(4(2(2(1(3(3(x1))))))) 2^1(0(3(0(2(x1))))) -> 4^1(2(2(1(3(3(x1)))))) 2^1(0(3(0(2(x1))))) -> 2^1(2(1(3(3(x1))))) 2^1(0(3(0(2(x1))))) -> 2^1(1(3(3(x1)))) 2^1(0(3(0(2(x1))))) -> 1^1(3(3(x1))) 2^1(0(3(0(2(x1))))) -> 3^1(3(x1)) 2^1(0(3(0(2(x1))))) -> 3^1(x1) 0^1(1(0(1(2(x1))))) -> 2^1(1(1(3(3(4(5(4(5(3(x1)))))))))) 0^1(1(0(1(2(x1))))) -> 1^1(1(3(3(4(5(4(5(3(x1))))))))) 0^1(1(0(1(2(x1))))) -> 1^1(3(3(4(5(4(5(3(x1)))))))) 0^1(1(0(1(2(x1))))) -> 3^1(3(4(5(4(5(3(x1))))))) 0^1(1(0(1(2(x1))))) -> 3^1(4(5(4(5(3(x1)))))) 0^1(1(0(1(2(x1))))) -> 4^1(5(4(5(3(x1))))) 0^1(1(0(1(2(x1))))) -> 5^1(4(5(3(x1)))) 0^1(1(0(1(2(x1))))) -> 4^1(5(3(x1))) 0^1(1(0(1(2(x1))))) -> 5^1(3(x1)) 0^1(1(0(1(2(x1))))) -> 3^1(x1) 2^1(0(2(4(3(x1))))) -> 2^1(3(5(2(3(3(0(3(5(3(x1)))))))))) 2^1(0(2(4(3(x1))))) -> 3^1(5(2(3(3(0(3(5(3(x1))))))))) 2^1(0(2(4(3(x1))))) -> 5^1(2(3(3(0(3(5(3(x1)))))))) 2^1(0(2(4(3(x1))))) -> 2^1(3(3(0(3(5(3(x1))))))) 2^1(0(2(4(3(x1))))) -> 3^1(3(0(3(5(3(x1)))))) 2^1(0(2(4(3(x1))))) -> 3^1(0(3(5(3(x1))))) 2^1(0(2(4(3(x1))))) -> 0^1(3(5(3(x1)))) 2^1(0(2(4(3(x1))))) -> 3^1(5(3(x1))) 2^1(0(2(4(3(x1))))) -> 5^1(3(x1)) 0^1(4(2(5(3(0(x1)))))) -> 2^1(3(5(2(2(3(2(0(4(4(x1)))))))))) 0^1(4(2(5(3(0(x1)))))) -> 3^1(5(2(2(3(2(0(4(4(x1))))))))) 0^1(4(2(5(3(0(x1)))))) -> 5^1(2(2(3(2(0(4(4(x1)))))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(2(3(2(0(4(4(x1))))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(3(2(0(4(4(x1)))))) 0^1(4(2(5(3(0(x1)))))) -> 3^1(2(0(4(4(x1))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(0(4(4(x1)))) 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) 0^1(4(2(5(3(0(x1)))))) -> 4^1(4(x1)) 0^1(4(2(5(3(0(x1)))))) -> 4^1(x1) 5^1(4(0(2(1(1(x1)))))) -> 3^1(3(2(0(1(2(4(5(0(3(x1)))))))))) 5^1(4(0(2(1(1(x1)))))) -> 3^1(2(0(1(2(4(5(0(3(x1))))))))) 5^1(4(0(2(1(1(x1)))))) -> 2^1(0(1(2(4(5(0(3(x1)))))))) 5^1(4(0(2(1(1(x1)))))) -> 0^1(1(2(4(5(0(3(x1))))))) 5^1(4(0(2(1(1(x1)))))) -> 1^1(2(4(5(0(3(x1)))))) 5^1(4(0(2(1(1(x1)))))) -> 2^1(4(5(0(3(x1))))) 5^1(4(0(2(1(1(x1)))))) -> 4^1(5(0(3(x1)))) 5^1(4(0(2(1(1(x1)))))) -> 5^1(0(3(x1))) 5^1(4(0(2(1(1(x1)))))) -> 0^1(3(x1)) 5^1(4(0(2(1(1(x1)))))) -> 3^1(x1) 2^1(1(0(1(1(2(x1)))))) -> 5^1(5(2(2(2(3(1(4(4(3(x1)))))))))) 2^1(1(0(1(1(2(x1)))))) -> 5^1(2(2(2(3(1(4(4(3(x1))))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(2(2(3(1(4(4(3(x1)))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(2(3(1(4(4(3(x1))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(3(1(4(4(3(x1)))))) 2^1(1(0(1(1(2(x1)))))) -> 3^1(1(4(4(3(x1))))) 2^1(1(0(1(1(2(x1)))))) -> 1^1(4(4(3(x1)))) 2^1(1(0(1(1(2(x1)))))) -> 4^1(4(3(x1))) 2^1(1(0(1(1(2(x1)))))) -> 4^1(3(x1)) 2^1(1(0(1(1(2(x1)))))) -> 3^1(x1) 1^1(1(1(0(2(2(x1)))))) -> 4^1(5(2(2(2(5(1(4(3(2(x1)))))))))) 1^1(1(1(0(2(2(x1)))))) -> 5^1(2(2(2(5(1(4(3(2(x1))))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(2(2(5(1(4(3(2(x1)))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(2(5(1(4(3(2(x1))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(5(1(4(3(2(x1)))))) 1^1(1(1(0(2(2(x1)))))) -> 5^1(1(4(3(2(x1))))) 1^1(1(1(0(2(2(x1)))))) -> 1^1(4(3(2(x1)))) 1^1(1(1(0(2(2(x1)))))) -> 4^1(3(2(x1))) 1^1(1(1(0(2(2(x1)))))) -> 3^1(2(x1)) 2^1(4(0(1(4(2(x1)))))) -> 0^1(4(4(4(3(2(3(1(5(1(x1)))))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(4(4(3(2(3(1(5(1(x1))))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(4(3(2(3(1(5(1(x1)))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(3(2(3(1(5(1(x1))))))) 2^1(4(0(1(4(2(x1)))))) -> 3^1(2(3(1(5(1(x1)))))) 2^1(4(0(1(4(2(x1)))))) -> 2^1(3(1(5(1(x1))))) 2^1(4(0(1(4(2(x1)))))) -> 3^1(1(5(1(x1)))) 2^1(4(0(1(4(2(x1)))))) -> 1^1(5(1(x1))) 2^1(4(0(1(4(2(x1)))))) -> 5^1(1(x1)) 2^1(4(0(1(4(2(x1)))))) -> 1^1(x1) 1^1(1(1(2(4(2(x1)))))) -> 4^1(4(1(3(4(3(4(5(3(1(x1)))))))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(1(3(4(3(4(5(3(1(x1))))))))) 1^1(1(1(2(4(2(x1)))))) -> 1^1(3(4(3(4(5(3(1(x1)))))))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(4(3(4(5(3(1(x1))))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(3(4(5(3(1(x1)))))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(4(5(3(1(x1))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(5(3(1(x1)))) 1^1(1(1(2(4(2(x1)))))) -> 5^1(3(1(x1))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(1(x1)) 1^1(1(1(2(4(2(x1)))))) -> 1^1(x1) 2^1(0(0(1(0(3(x1)))))) -> 2^1(3(3(0(5(3(5(2(4(2(x1)))))))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(3(0(5(3(5(2(4(2(x1))))))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(0(5(3(5(2(4(2(x1)))))))) 2^1(0(0(1(0(3(x1)))))) -> 0^1(5(3(5(2(4(2(x1))))))) 2^1(0(0(1(0(3(x1)))))) -> 5^1(3(5(2(4(2(x1)))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(5(2(4(2(x1))))) 2^1(0(0(1(0(3(x1)))))) -> 5^1(2(4(2(x1)))) 2^1(0(0(1(0(3(x1)))))) -> 2^1(4(2(x1))) 2^1(0(0(1(0(3(x1)))))) -> 4^1(2(x1)) 2^1(0(0(1(0(3(x1)))))) -> 2^1(x1) 1^1(1(1(1(0(3(x1)))))) -> 1^1(5(4(2(5(4(4(2(2(3(x1)))))))))) 1^1(1(1(1(0(3(x1)))))) -> 5^1(4(2(5(4(4(2(2(3(x1))))))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(2(5(4(4(2(2(3(x1)))))))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(5(4(4(2(2(3(x1))))))) 1^1(1(1(1(0(3(x1)))))) -> 5^1(4(4(2(2(3(x1)))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(4(2(2(3(x1))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(2(2(3(x1)))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(2(3(x1))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(3(x1)) 2^1(0(2(1(1(4(x1)))))) -> 3^1(2(3(2(4(4(4(3(0(4(x1)))))))))) 2^1(0(2(1(1(4(x1)))))) -> 2^1(3(2(4(4(4(3(0(4(x1))))))))) 2^1(0(2(1(1(4(x1)))))) -> 3^1(2(4(4(4(3(0(4(x1)))))))) 2^1(0(2(1(1(4(x1)))))) -> 2^1(4(4(4(3(0(4(x1))))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(4(4(3(0(4(x1)))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(4(3(0(4(x1))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(3(0(4(x1)))) 2^1(0(2(1(1(4(x1)))))) -> 3^1(0(4(x1))) 2^1(0(2(1(1(4(x1)))))) -> 0^1(4(x1)) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(5(2(5(3(5(5(1(0(x1)))))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(2(5(3(5(5(1(0(x1))))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(2(5(3(5(5(1(0(x1)))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 2^1(5(3(5(5(1(0(x1))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(3(5(5(1(0(x1)))))) 0^1(1(1(1(1(2(0(x1))))))) -> 3^1(5(5(1(0(x1))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(1(0(x1)))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(1(0(x1))) 0^1(1(1(1(1(2(0(x1))))))) -> 1^1(0(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(2(0(3(2(3(4(3(4(4(x1)))))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(0(3(2(3(4(3(4(4(x1))))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 0^1(3(2(3(4(3(4(4(x1)))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(2(3(4(3(4(4(x1))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(3(4(3(4(4(x1)))))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(4(3(4(4(x1))))) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(3(4(4(x1)))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(4(4(x1))) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(4(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(x1) 1^1(4(0(0(2(4(0(x1))))))) -> 1^1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 3^1(3(4(0(1(4(5(2(4(x1))))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 3^1(4(0(1(4(5(2(4(x1)))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(0(1(4(5(2(4(x1))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 0^1(1(4(5(2(4(x1)))))) 1^1(4(0(0(2(4(0(x1))))))) -> 1^1(4(5(2(4(x1))))) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(5(2(4(x1)))) 1^1(4(0(0(2(4(0(x1))))))) -> 5^1(2(4(x1))) 1^1(4(0(0(2(4(0(x1))))))) -> 2^1(4(x1)) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(x1) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(4(5(2(1(3(5(1(3(0(x1)))))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 4^1(5(2(1(3(5(1(3(0(x1))))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 5^1(2(1(3(5(1(3(0(x1)))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 2^1(1(3(5(1(3(0(x1))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(3(5(1(3(0(x1)))))) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(5(1(3(0(x1))))) 1^1(4(5(0(3(4(0(x1))))))) -> 5^1(1(3(0(x1)))) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(3(0(x1))) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(0(x1)) 0^1(0(2(2(5(0(1(x1))))))) -> 2^1(5(4(5(4(3(4(4(5(1(x1)))))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(4(5(4(3(4(4(5(1(x1))))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(5(4(3(4(4(5(1(x1)))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(4(3(4(4(5(1(x1))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(3(4(4(5(1(x1)))))) 0^1(0(2(2(5(0(1(x1))))))) -> 3^1(4(4(5(1(x1))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(4(5(1(x1)))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(5(1(x1))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(1(x1)) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(5(4(1(4(3(5(1(3(1(x1)))))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 5^1(4(1(4(3(5(1(3(1(x1))))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 4^1(1(4(3(5(1(3(1(x1)))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 1^1(4(3(5(1(3(1(x1))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 4^1(3(5(1(3(1(x1)))))) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(5(1(3(1(x1))))) 0^1(0(5(4(3(1(1(x1))))))) -> 5^1(1(3(1(x1)))) 0^1(0(5(4(3(1(1(x1))))))) -> 1^1(3(1(x1))) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(1(x1)) 5^1(0(5(1(3(4(1(x1))))))) -> 2^1(3(3(1(4(2(3(3(0(5(x1)))))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(3(1(4(2(3(3(0(5(x1))))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(1(4(2(3(3(0(5(x1)))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 1^1(4(2(3(3(0(5(x1))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 4^1(2(3(3(0(5(x1)))))) 5^1(0(5(1(3(4(1(x1))))))) -> 2^1(3(3(0(5(x1))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(3(0(5(x1)))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(0(5(x1))) 5^1(0(5(1(3(4(1(x1))))))) -> 0^1(5(x1)) 5^1(0(5(1(3(4(1(x1))))))) -> 5^1(x1) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(5(0(0(2(4(5(2(5(2(x1)))))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(0(0(2(4(5(2(5(2(x1))))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 0^1(0(2(4(5(2(5(2(x1)))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 0^1(2(4(5(2(5(2(x1))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(4(5(2(5(2(x1)))))) 5^1(5(0(2(0(5(1(x1))))))) -> 4^1(5(2(5(2(x1))))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(2(5(2(x1)))) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(5(2(x1))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(2(x1)) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(x1) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(5(3(1(3(5(5(3(3(4(x1)))))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(3(1(3(5(5(3(3(4(x1))))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(1(3(5(5(3(3(4(x1)))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 1^1(3(5(5(3(3(4(x1))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(5(5(3(3(4(x1)))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(5(3(3(4(x1))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(3(3(4(x1)))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(3(4(x1))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(4(x1)) 5^1(0(2(5(1(0(2(x1))))))) -> 4^1(x1) 4^1(1(4(5(0(4(2(x1))))))) -> 4^1(1(5(3(5(1(5(5(4(3(x1)))))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 1^1(5(3(5(1(5(5(4(3(x1))))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(3(5(1(5(5(4(3(x1)))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(5(1(5(5(4(3(x1))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(1(5(5(4(3(x1)))))) 4^1(1(4(5(0(4(2(x1))))))) -> 1^1(5(5(4(3(x1))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(5(4(3(x1)))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(4(3(x1))) 4^1(1(4(5(0(4(2(x1))))))) -> 4^1(3(x1)) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(x1) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(0(3(3(4(3(1(2(2(3(x1)))))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 0^1(3(3(4(3(1(2(2(3(x1))))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(3(4(3(1(2(2(3(x1)))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(4(3(1(2(2(3(x1))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 4^1(3(1(2(2(3(x1)))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(1(2(2(3(x1))))) 5^1(4(0(4(1(4(3(x1))))))) -> 1^1(2(2(3(x1)))) 5^1(4(0(4(1(4(3(x1))))))) -> 2^1(2(3(x1))) 5^1(4(0(4(1(4(3(x1))))))) -> 2^1(3(x1)) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(4(3(0(0(1(3(2(2(4(x1)))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 4^1(3(0(0(1(3(2(2(4(x1))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(0(0(1(3(2(2(4(x1)))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(1(3(2(2(4(x1))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(1(3(2(2(4(x1)))))) 0^1(0(2(4(0(1(4(x1))))))) -> 1^1(3(2(2(4(x1))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(2(2(4(x1)))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(2(4(x1))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(4(x1)) 3^1(0(2(4(0(1(4(x1))))))) -> 2^1(3(4(5(1(0(0(3(4(0(x1)))))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 3^1(4(5(1(0(0(3(4(0(x1))))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 4^1(5(1(0(0(3(4(0(x1)))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 5^1(1(0(0(3(4(0(x1))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 1^1(0(0(3(4(0(x1)))))) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(3(4(0(x1))))) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(3(4(0(x1)))) 3^1(0(2(4(0(1(4(x1))))))) -> 3^1(4(0(x1))) 3^1(0(2(4(0(1(4(x1))))))) -> 4^1(0(x1)) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(x1) 2^1(1(0(1(1(1(4(x1))))))) -> 2^1(5(5(1(0(3(3(2(3(3(x1)))))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 5^1(5(1(0(3(3(2(3(3(x1))))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 5^1(1(0(3(3(2(3(3(x1)))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 1^1(0(3(3(2(3(3(x1))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 0^1(3(3(2(3(3(x1)))))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(3(2(3(3(x1))))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(2(3(3(x1)))) 2^1(1(0(1(1(1(4(x1))))))) -> 2^1(3(3(x1))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(3(x1)) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(x1) The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) SplitQDPProof (EQUIVALENT) We show in the first subproof that some pairs and rules can be removed, afterwards, we continue with the remaining DP-Problem ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(2(1(x1))) -> 3^1(4(1(5(4(5(3(3(0(4(x1)))))))))) 0^1(2(1(x1))) -> 4^1(1(5(4(5(3(3(0(4(x1))))))))) 0^1(2(1(x1))) -> 1^1(5(4(5(3(3(0(4(x1)))))))) 0^1(2(1(x1))) -> 5^1(4(5(3(3(0(4(x1))))))) 0^1(2(1(x1))) -> 4^1(5(3(3(0(4(x1)))))) 0^1(2(1(x1))) -> 5^1(3(3(0(4(x1))))) 0^1(2(1(x1))) -> 3^1(3(0(4(x1)))) 0^1(2(1(x1))) -> 3^1(0(4(x1))) 0^1(2(1(x1))) -> 0^1(4(x1)) 0^1(2(1(x1))) -> 4^1(x1) 5^1(4(0(0(1(x1))))) -> 3^1(2(5(4(1(3(1(3(4(1(x1)))))))))) 5^1(4(0(0(1(x1))))) -> 2^1(5(4(1(3(1(3(4(1(x1))))))))) 5^1(4(0(0(1(x1))))) -> 5^1(4(1(3(1(3(4(1(x1)))))))) 5^1(4(0(0(1(x1))))) -> 4^1(1(3(1(3(4(1(x1))))))) 5^1(4(0(0(1(x1))))) -> 1^1(3(1(3(4(1(x1)))))) 5^1(4(0(0(1(x1))))) -> 3^1(1(3(4(1(x1))))) 5^1(4(0(0(1(x1))))) -> 1^1(3(4(1(x1)))) 5^1(4(0(0(1(x1))))) -> 3^1(4(1(x1))) 5^1(4(0(0(1(x1))))) -> 4^1(1(x1)) 2^1(0(3(0(2(x1))))) -> 3^1(4(0(5(4(2(2(1(3(3(x1)))))))))) 2^1(0(3(0(2(x1))))) -> 4^1(0(5(4(2(2(1(3(3(x1))))))))) 2^1(0(3(0(2(x1))))) -> 0^1(5(4(2(2(1(3(3(x1)))))))) 2^1(0(3(0(2(x1))))) -> 5^1(4(2(2(1(3(3(x1))))))) 2^1(0(3(0(2(x1))))) -> 4^1(2(2(1(3(3(x1)))))) 2^1(0(3(0(2(x1))))) -> 2^1(2(1(3(3(x1))))) 2^1(0(3(0(2(x1))))) -> 2^1(1(3(3(x1)))) 2^1(0(3(0(2(x1))))) -> 1^1(3(3(x1))) 2^1(0(3(0(2(x1))))) -> 3^1(3(x1)) 2^1(0(3(0(2(x1))))) -> 3^1(x1) 0^1(1(0(1(2(x1))))) -> 2^1(1(1(3(3(4(5(4(5(3(x1)))))))))) 0^1(1(0(1(2(x1))))) -> 1^1(1(3(3(4(5(4(5(3(x1))))))))) 0^1(1(0(1(2(x1))))) -> 1^1(3(3(4(5(4(5(3(x1)))))))) 0^1(1(0(1(2(x1))))) -> 3^1(3(4(5(4(5(3(x1))))))) 0^1(1(0(1(2(x1))))) -> 3^1(4(5(4(5(3(x1)))))) 0^1(1(0(1(2(x1))))) -> 4^1(5(4(5(3(x1))))) 0^1(1(0(1(2(x1))))) -> 5^1(4(5(3(x1)))) 0^1(1(0(1(2(x1))))) -> 4^1(5(3(x1))) 0^1(1(0(1(2(x1))))) -> 5^1(3(x1)) 0^1(1(0(1(2(x1))))) -> 3^1(x1) 2^1(0(2(4(3(x1))))) -> 2^1(3(5(2(3(3(0(3(5(3(x1)))))))))) 2^1(0(2(4(3(x1))))) -> 3^1(5(2(3(3(0(3(5(3(x1))))))))) 2^1(0(2(4(3(x1))))) -> 5^1(2(3(3(0(3(5(3(x1)))))))) 2^1(0(2(4(3(x1))))) -> 2^1(3(3(0(3(5(3(x1))))))) 2^1(0(2(4(3(x1))))) -> 3^1(3(0(3(5(3(x1)))))) 2^1(0(2(4(3(x1))))) -> 3^1(0(3(5(3(x1))))) 2^1(0(2(4(3(x1))))) -> 0^1(3(5(3(x1)))) 2^1(0(2(4(3(x1))))) -> 3^1(5(3(x1))) 2^1(0(2(4(3(x1))))) -> 5^1(3(x1)) 0^1(4(2(5(3(0(x1)))))) -> 2^1(3(5(2(2(3(2(0(4(4(x1)))))))))) 0^1(4(2(5(3(0(x1)))))) -> 3^1(5(2(2(3(2(0(4(4(x1))))))))) 0^1(4(2(5(3(0(x1)))))) -> 5^1(2(2(3(2(0(4(4(x1)))))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(2(3(2(0(4(4(x1))))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(3(2(0(4(4(x1)))))) 0^1(4(2(5(3(0(x1)))))) -> 3^1(2(0(4(4(x1))))) 0^1(4(2(5(3(0(x1)))))) -> 2^1(0(4(4(x1)))) 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) 0^1(4(2(5(3(0(x1)))))) -> 4^1(4(x1)) 0^1(4(2(5(3(0(x1)))))) -> 4^1(x1) 5^1(4(0(2(1(1(x1)))))) -> 3^1(3(2(0(1(2(4(5(0(3(x1)))))))))) 5^1(4(0(2(1(1(x1)))))) -> 3^1(2(0(1(2(4(5(0(3(x1))))))))) 5^1(4(0(2(1(1(x1)))))) -> 2^1(0(1(2(4(5(0(3(x1)))))))) 5^1(4(0(2(1(1(x1)))))) -> 0^1(1(2(4(5(0(3(x1))))))) 5^1(4(0(2(1(1(x1)))))) -> 1^1(2(4(5(0(3(x1)))))) 5^1(4(0(2(1(1(x1)))))) -> 2^1(4(5(0(3(x1))))) 5^1(4(0(2(1(1(x1)))))) -> 4^1(5(0(3(x1)))) 5^1(4(0(2(1(1(x1)))))) -> 5^1(0(3(x1))) 5^1(4(0(2(1(1(x1)))))) -> 0^1(3(x1)) 5^1(4(0(2(1(1(x1)))))) -> 3^1(x1) 2^1(1(0(1(1(2(x1)))))) -> 5^1(5(2(2(2(3(1(4(4(3(x1)))))))))) 2^1(1(0(1(1(2(x1)))))) -> 5^1(2(2(2(3(1(4(4(3(x1))))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(2(2(3(1(4(4(3(x1)))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(2(3(1(4(4(3(x1))))))) 2^1(1(0(1(1(2(x1)))))) -> 2^1(3(1(4(4(3(x1)))))) 2^1(1(0(1(1(2(x1)))))) -> 3^1(1(4(4(3(x1))))) 2^1(1(0(1(1(2(x1)))))) -> 1^1(4(4(3(x1)))) 2^1(1(0(1(1(2(x1)))))) -> 4^1(4(3(x1))) 2^1(1(0(1(1(2(x1)))))) -> 4^1(3(x1)) 2^1(1(0(1(1(2(x1)))))) -> 3^1(x1) 1^1(1(1(0(2(2(x1)))))) -> 4^1(5(2(2(2(5(1(4(3(2(x1)))))))))) 1^1(1(1(0(2(2(x1)))))) -> 5^1(2(2(2(5(1(4(3(2(x1))))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(2(2(5(1(4(3(2(x1)))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(2(5(1(4(3(2(x1))))))) 1^1(1(1(0(2(2(x1)))))) -> 2^1(5(1(4(3(2(x1)))))) 1^1(1(1(0(2(2(x1)))))) -> 5^1(1(4(3(2(x1))))) 1^1(1(1(0(2(2(x1)))))) -> 1^1(4(3(2(x1)))) 1^1(1(1(0(2(2(x1)))))) -> 4^1(3(2(x1))) 1^1(1(1(0(2(2(x1)))))) -> 3^1(2(x1)) 2^1(4(0(1(4(2(x1)))))) -> 0^1(4(4(4(3(2(3(1(5(1(x1)))))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(4(4(3(2(3(1(5(1(x1))))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(4(3(2(3(1(5(1(x1)))))))) 2^1(4(0(1(4(2(x1)))))) -> 4^1(3(2(3(1(5(1(x1))))))) 2^1(4(0(1(4(2(x1)))))) -> 3^1(2(3(1(5(1(x1)))))) 2^1(4(0(1(4(2(x1)))))) -> 2^1(3(1(5(1(x1))))) 2^1(4(0(1(4(2(x1)))))) -> 3^1(1(5(1(x1)))) 2^1(4(0(1(4(2(x1)))))) -> 1^1(5(1(x1))) 2^1(4(0(1(4(2(x1)))))) -> 5^1(1(x1)) 2^1(4(0(1(4(2(x1)))))) -> 1^1(x1) 1^1(1(1(2(4(2(x1)))))) -> 4^1(4(1(3(4(3(4(5(3(1(x1)))))))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(1(3(4(3(4(5(3(1(x1))))))))) 1^1(1(1(2(4(2(x1)))))) -> 1^1(3(4(3(4(5(3(1(x1)))))))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(4(3(4(5(3(1(x1))))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(3(4(5(3(1(x1)))))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(4(5(3(1(x1))))) 1^1(1(1(2(4(2(x1)))))) -> 4^1(5(3(1(x1)))) 1^1(1(1(2(4(2(x1)))))) -> 5^1(3(1(x1))) 1^1(1(1(2(4(2(x1)))))) -> 3^1(1(x1)) 1^1(1(1(2(4(2(x1)))))) -> 1^1(x1) 2^1(0(0(1(0(3(x1)))))) -> 2^1(3(3(0(5(3(5(2(4(2(x1)))))))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(3(0(5(3(5(2(4(2(x1))))))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(0(5(3(5(2(4(2(x1)))))))) 2^1(0(0(1(0(3(x1)))))) -> 0^1(5(3(5(2(4(2(x1))))))) 2^1(0(0(1(0(3(x1)))))) -> 5^1(3(5(2(4(2(x1)))))) 2^1(0(0(1(0(3(x1)))))) -> 3^1(5(2(4(2(x1))))) 2^1(0(0(1(0(3(x1)))))) -> 5^1(2(4(2(x1)))) 2^1(0(0(1(0(3(x1)))))) -> 2^1(4(2(x1))) 2^1(0(0(1(0(3(x1)))))) -> 4^1(2(x1)) 2^1(0(0(1(0(3(x1)))))) -> 2^1(x1) 1^1(1(1(1(0(3(x1)))))) -> 1^1(5(4(2(5(4(4(2(2(3(x1)))))))))) 1^1(1(1(1(0(3(x1)))))) -> 5^1(4(2(5(4(4(2(2(3(x1))))))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(2(5(4(4(2(2(3(x1)))))))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(5(4(4(2(2(3(x1))))))) 1^1(1(1(1(0(3(x1)))))) -> 5^1(4(4(2(2(3(x1)))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(4(2(2(3(x1))))) 1^1(1(1(1(0(3(x1)))))) -> 4^1(2(2(3(x1)))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(2(3(x1))) 1^1(1(1(1(0(3(x1)))))) -> 2^1(3(x1)) 2^1(0(2(1(1(4(x1)))))) -> 3^1(2(3(2(4(4(4(3(0(4(x1)))))))))) 2^1(0(2(1(1(4(x1)))))) -> 2^1(3(2(4(4(4(3(0(4(x1))))))))) 2^1(0(2(1(1(4(x1)))))) -> 3^1(2(4(4(4(3(0(4(x1)))))))) 2^1(0(2(1(1(4(x1)))))) -> 2^1(4(4(4(3(0(4(x1))))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(4(4(3(0(4(x1)))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(4(3(0(4(x1))))) 2^1(0(2(1(1(4(x1)))))) -> 4^1(3(0(4(x1)))) 2^1(0(2(1(1(4(x1)))))) -> 3^1(0(4(x1))) 2^1(0(2(1(1(4(x1)))))) -> 0^1(4(x1)) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(5(2(5(3(5(5(1(0(x1)))))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(2(5(3(5(5(1(0(x1))))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(2(5(3(5(5(1(0(x1)))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 2^1(5(3(5(5(1(0(x1))))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(3(5(5(1(0(x1)))))) 0^1(1(1(1(1(2(0(x1))))))) -> 3^1(5(5(1(0(x1))))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(5(1(0(x1)))) 0^1(1(1(1(1(2(0(x1))))))) -> 5^1(1(0(x1))) 0^1(1(1(1(1(2(0(x1))))))) -> 1^1(0(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(2(0(3(2(3(4(3(4(4(x1)))))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(0(3(2(3(4(3(4(4(x1))))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 0^1(3(2(3(4(3(4(4(x1)))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(2(3(4(3(4(4(x1))))))) 5^1(1(1(1(4(2(0(x1))))))) -> 2^1(3(4(3(4(4(x1)))))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(4(3(4(4(x1))))) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(3(4(4(x1)))) 5^1(1(1(1(4(2(0(x1))))))) -> 3^1(4(4(x1))) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(4(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(x1) 1^1(4(0(0(2(4(0(x1))))))) -> 1^1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 3^1(3(4(0(1(4(5(2(4(x1))))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 3^1(4(0(1(4(5(2(4(x1)))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(0(1(4(5(2(4(x1))))))) 1^1(4(0(0(2(4(0(x1))))))) -> 0^1(1(4(5(2(4(x1)))))) 1^1(4(0(0(2(4(0(x1))))))) -> 1^1(4(5(2(4(x1))))) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(5(2(4(x1)))) 1^1(4(0(0(2(4(0(x1))))))) -> 5^1(2(4(x1))) 1^1(4(0(0(2(4(0(x1))))))) -> 2^1(4(x1)) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(x1) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(4(5(2(1(3(5(1(3(0(x1)))))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 4^1(5(2(1(3(5(1(3(0(x1))))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 5^1(2(1(3(5(1(3(0(x1)))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 2^1(1(3(5(1(3(0(x1))))))) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(3(5(1(3(0(x1)))))) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(5(1(3(0(x1))))) 1^1(4(5(0(3(4(0(x1))))))) -> 5^1(1(3(0(x1)))) 1^1(4(5(0(3(4(0(x1))))))) -> 1^1(3(0(x1))) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(0(x1)) 0^1(0(2(2(5(0(1(x1))))))) -> 2^1(5(4(5(4(3(4(4(5(1(x1)))))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(4(5(4(3(4(4(5(1(x1))))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(5(4(3(4(4(5(1(x1)))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(4(3(4(4(5(1(x1))))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(3(4(4(5(1(x1)))))) 0^1(0(2(2(5(0(1(x1))))))) -> 3^1(4(4(5(1(x1))))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(4(5(1(x1)))) 0^1(0(2(2(5(0(1(x1))))))) -> 4^1(5(1(x1))) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(1(x1)) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(5(4(1(4(3(5(1(3(1(x1)))))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 5^1(4(1(4(3(5(1(3(1(x1))))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 4^1(1(4(3(5(1(3(1(x1)))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 1^1(4(3(5(1(3(1(x1))))))) 0^1(0(5(4(3(1(1(x1))))))) -> 4^1(3(5(1(3(1(x1)))))) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(5(1(3(1(x1))))) 0^1(0(5(4(3(1(1(x1))))))) -> 5^1(1(3(1(x1)))) 0^1(0(5(4(3(1(1(x1))))))) -> 1^1(3(1(x1))) 0^1(0(5(4(3(1(1(x1))))))) -> 3^1(1(x1)) 5^1(0(5(1(3(4(1(x1))))))) -> 2^1(3(3(1(4(2(3(3(0(5(x1)))))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(3(1(4(2(3(3(0(5(x1))))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(1(4(2(3(3(0(5(x1)))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 1^1(4(2(3(3(0(5(x1))))))) 5^1(0(5(1(3(4(1(x1))))))) -> 4^1(2(3(3(0(5(x1)))))) 5^1(0(5(1(3(4(1(x1))))))) -> 2^1(3(3(0(5(x1))))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(3(0(5(x1)))) 5^1(0(5(1(3(4(1(x1))))))) -> 3^1(0(5(x1))) 5^1(0(5(1(3(4(1(x1))))))) -> 0^1(5(x1)) 5^1(0(5(1(3(4(1(x1))))))) -> 5^1(x1) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(5(0(0(2(4(5(2(5(2(x1)))))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(0(0(2(4(5(2(5(2(x1))))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 0^1(0(2(4(5(2(5(2(x1)))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 0^1(2(4(5(2(5(2(x1))))))) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(4(5(2(5(2(x1)))))) 5^1(5(0(2(0(5(1(x1))))))) -> 4^1(5(2(5(2(x1))))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(2(5(2(x1)))) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(5(2(x1))) 5^1(5(0(2(0(5(1(x1))))))) -> 5^1(2(x1)) 5^1(5(0(2(0(5(1(x1))))))) -> 2^1(x1) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(5(3(1(3(5(5(3(3(4(x1)))))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(3(1(3(5(5(3(3(4(x1))))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(1(3(5(5(3(3(4(x1)))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 1^1(3(5(5(3(3(4(x1))))))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(5(5(3(3(4(x1)))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(5(3(3(4(x1))))) 5^1(0(2(5(1(0(2(x1))))))) -> 5^1(3(3(4(x1)))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(3(4(x1))) 5^1(0(2(5(1(0(2(x1))))))) -> 3^1(4(x1)) 5^1(0(2(5(1(0(2(x1))))))) -> 4^1(x1) 4^1(1(4(5(0(4(2(x1))))))) -> 4^1(1(5(3(5(1(5(5(4(3(x1)))))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 1^1(5(3(5(1(5(5(4(3(x1))))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(3(5(1(5(5(4(3(x1)))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(5(1(5(5(4(3(x1))))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(1(5(5(4(3(x1)))))) 4^1(1(4(5(0(4(2(x1))))))) -> 1^1(5(5(4(3(x1))))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(5(4(3(x1)))) 4^1(1(4(5(0(4(2(x1))))))) -> 5^1(4(3(x1))) 4^1(1(4(5(0(4(2(x1))))))) -> 4^1(3(x1)) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(x1) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(0(3(3(4(3(1(2(2(3(x1)))))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 0^1(3(3(4(3(1(2(2(3(x1))))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(3(4(3(1(2(2(3(x1)))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(4(3(1(2(2(3(x1))))))) 5^1(4(0(4(1(4(3(x1))))))) -> 4^1(3(1(2(2(3(x1)))))) 5^1(4(0(4(1(4(3(x1))))))) -> 3^1(1(2(2(3(x1))))) 5^1(4(0(4(1(4(3(x1))))))) -> 1^1(2(2(3(x1)))) 5^1(4(0(4(1(4(3(x1))))))) -> 2^1(2(3(x1))) 5^1(4(0(4(1(4(3(x1))))))) -> 2^1(3(x1)) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(4(3(0(0(1(3(2(2(4(x1)))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 4^1(3(0(0(1(3(2(2(4(x1))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(0(0(1(3(2(2(4(x1)))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(1(3(2(2(4(x1))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(1(3(2(2(4(x1)))))) 0^1(0(2(4(0(1(4(x1))))))) -> 1^1(3(2(2(4(x1))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(2(2(4(x1)))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(2(4(x1))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(4(x1)) 3^1(0(2(4(0(1(4(x1))))))) -> 2^1(3(4(5(1(0(0(3(4(0(x1)))))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 3^1(4(5(1(0(0(3(4(0(x1))))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 4^1(5(1(0(0(3(4(0(x1)))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 5^1(1(0(0(3(4(0(x1))))))) 3^1(0(2(4(0(1(4(x1))))))) -> 1^1(0(0(3(4(0(x1)))))) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(3(4(0(x1))))) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(3(4(0(x1)))) 3^1(0(2(4(0(1(4(x1))))))) -> 3^1(4(0(x1))) 3^1(0(2(4(0(1(4(x1))))))) -> 4^1(0(x1)) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(x1) 2^1(1(0(1(1(1(4(x1))))))) -> 2^1(5(5(1(0(3(3(2(3(3(x1)))))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 5^1(5(1(0(3(3(2(3(3(x1))))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 5^1(1(0(3(3(2(3(3(x1)))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 1^1(0(3(3(2(3(3(x1))))))) 2^1(1(0(1(1(1(4(x1))))))) -> 0^1(3(3(2(3(3(x1)))))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(3(2(3(3(x1))))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(2(3(3(x1)))) 2^1(1(0(1(1(1(4(x1))))))) -> 2^1(3(3(x1))) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(3(x1)) 2^1(1(0(1(1(1(4(x1))))))) -> 3^1(x1) The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) SemLabProof (SOUND) We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1. 5: 0 3^1: 0 0^1: 0 4^1: 0 1^1: 0 5^1: 0 2^1: 0 0: 0 1: 1 2: 0 3: 0 4: 1 By semantic labelling [SEMLAB] we obtain the following labelled QDP problem. ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1.0(2.1(1.0(x1))) -> 3^1.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 0^1.0(2.1(1.1(x1))) -> 3^1.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1))))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1))))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 4^1.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 4^1.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 5^1.1(1.0(0.0(0.0(3.1(4.0(0.0(x1))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 5^1.1(1.0(0.0(0.0(3.1(4.0(0.1(x1))))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 1^1.0(0.0(0.0(3.1(4.0(0.0(x1)))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 1^1.0(0.0(0.0(3.1(4.0(0.1(x1)))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(0.0(3.1(4.0(0.0(x1))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(0.0(3.1(4.0(0.1(x1))))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(3.1(4.0(0.0(x1)))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(3.1(4.0(0.1(x1)))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.1(4.0(0.0(x1))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.1(4.0(0.1(x1))) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 4^1.0(0.0(x1)) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 4^1.0(0.1(x1)) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(x1) 0^1.0(2.1(1.0(x1))) -> 4^1.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1))))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4^1.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4^1.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 0^1.0(2.1(1.1(x1))) -> 4^1.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1))))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 1^1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1))))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 1^1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1))))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 5^1.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 5^1.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 3^1.0(5.1(1.0(5.0(5.1(4.0(3.0(x1))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 3^1.0(5.1(1.0(5.0(5.1(4.0(3.1(x1))))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 5^1.1(1.0(5.0(5.1(4.0(3.0(x1)))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 5^1.1(1.0(5.0(5.1(4.0(3.1(x1)))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 1^1.0(5.0(5.1(4.0(3.0(x1))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 1^1.0(5.0(5.1(4.0(3.1(x1))))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 5^1.0(5.1(4.0(3.0(x1)))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 5^1.0(5.1(4.0(3.1(x1)))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 5^1.1(4.0(3.0(x1))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 5^1.1(4.0(3.1(x1))) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4^1.0(3.0(x1)) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4^1.0(3.1(x1)) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 3^1.0(x1) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 3^1.1(x1) 0^1.0(2.1(1.0(x1))) -> 1^1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4^1.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4^1.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 0^1.0(2.1(1.1(x1))) -> 1^1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 5^1.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1))))))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 5^1.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1))))))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 2^1.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 2^1.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 2^1.0(2.0(5.1(1.1(4.0(3.0(2.0(x1))))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 2^1.0(2.0(5.1(1.1(4.0(3.0(2.1(x1))))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 2^1.0(5.1(1.1(4.0(3.0(2.0(x1)))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 2^1.0(5.1(1.1(4.0(3.0(2.1(x1)))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.0(x1))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.1(x1))))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 1^1.1(4.0(3.0(2.0(x1)))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 1^1.1(4.0(3.0(2.1(x1)))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4^1.0(3.0(2.0(x1))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4^1.0(3.0(2.1(x1))) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 3^1.0(2.0(x1)) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 3^1.0(2.1(x1)) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4^1.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4^1.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4^1.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1))))))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4^1.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1))))))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 1^1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 1^1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 3^1.1(4.0(3.1(4.0(5.0(3.1(1.0(x1))))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 3^1.1(4.0(3.1(4.0(5.0(3.1(1.1(x1))))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4^1.0(3.1(4.0(5.0(3.1(1.0(x1)))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4^1.0(3.1(4.0(5.0(3.1(1.1(x1)))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 3^1.1(4.0(5.0(3.1(1.0(x1))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 3^1.1(4.0(5.0(3.1(1.1(x1))))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4^1.0(5.0(3.1(1.0(x1)))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4^1.0(5.0(3.1(1.1(x1)))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 5^1.0(3.1(1.0(x1))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 5^1.0(3.1(1.1(x1))) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 3^1.1(1.0(x1)) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 3^1.1(1.1(x1)) 1^1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 1^1.0(x1) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1^1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1^1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 5^1.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1))))))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 5^1.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1))))))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 4^1.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 4^1.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(5.1(4.1(4.0(2.0(2.0(3.0(x1))))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 2^1.0(5.1(4.1(4.0(2.0(2.0(3.1(x1))))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 5^1.1(4.1(4.0(2.0(2.0(3.0(x1)))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 5^1.1(4.1(4.0(2.0(2.0(3.1(x1)))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 4^1.1(4.0(2.0(2.0(3.0(x1))))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 4^1.1(4.0(2.0(2.0(3.1(x1))))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 4^1.0(2.0(2.0(3.0(x1)))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 4^1.0(2.0(2.0(3.1(x1)))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(2.0(3.0(x1))) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 2^1.0(2.0(3.1(x1))) 1^1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(3.0(x1)) 1^1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 2^1.0(3.1(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1^1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1^1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 3^1.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1))))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 3^1.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1))))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 3^1.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 3^1.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 4^1.0(0.1(1.1(4.0(5.0(2.1(4.0(x1))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.0(0.1(1.1(4.0(5.0(2.1(4.1(x1))))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.0(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.1(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1^1.1(4.0(5.0(2.1(4.0(x1))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1^1.1(4.0(5.0(2.1(4.1(x1))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 4^1.0(5.0(2.1(4.0(x1)))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.0(5.0(2.1(4.1(x1)))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 5^1.0(2.1(4.0(x1))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 5^1.0(2.1(4.1(x1))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 2^1.1(4.0(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 2^1.1(4.1(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 4^1.0(x1) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.1(x1) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1^1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1^1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 4^1.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1))))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 4^1.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1))))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 5^1.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 5^1.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 2^1.1(1.0(3.0(5.1(1.0(3.0(0.0(x1))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 2^1.1(1.0(3.0(5.1(1.0(3.0(0.1(x1))))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1^1.0(3.0(5.1(1.0(3.0(0.0(x1)))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1^1.0(3.0(5.1(1.0(3.0(0.1(x1)))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 3^1.0(5.1(1.0(3.0(0.0(x1))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 3^1.0(5.1(1.0(3.0(0.1(x1))))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 5^1.1(1.0(3.0(0.0(x1)))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 5^1.1(1.0(3.0(0.1(x1)))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1^1.0(3.0(0.0(x1))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1^1.0(3.0(0.1(x1))) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 3^1.0(0.0(x1)) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 3^1.0(0.1(x1)) 0^1.0(2.1(1.0(x1))) -> 5^1.1(4.0(5.0(3.0(3.0(0.1(4.0(x1))))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 3^1.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 3^1.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 0^1.0(2.1(1.1(x1))) -> 5^1.1(4.0(5.0(3.0(3.0(0.1(4.1(x1))))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 2^1.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1))))))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 2^1.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1))))))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 5^1.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 5^1.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 4^1.1(1.0(3.1(1.0(3.1(4.1(1.0(x1))))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 4^1.1(1.0(3.1(1.0(3.1(4.1(1.1(x1))))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 1^1.0(3.1(1.0(3.1(4.1(1.0(x1)))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 1^1.0(3.1(1.0(3.1(4.1(1.1(x1)))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 3^1.1(1.0(3.1(4.1(1.0(x1))))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 3^1.1(1.0(3.1(4.1(1.1(x1))))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 1^1.0(3.1(4.1(1.0(x1)))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 1^1.0(3.1(4.1(1.1(x1)))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 3^1.1(4.1(1.0(x1))) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 3^1.1(4.1(1.1(x1))) 5^1.1(4.0(0.0(0.1(1.0(x1))))) -> 4^1.1(1.0(x1)) 5^1.1(4.0(0.0(0.1(1.1(x1))))) -> 4^1.1(1.1(x1)) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3^1.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3^1.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3^1.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1))))))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3^1.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1))))))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 2^1.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 2^1.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 0^1.1(1.0(2.1(4.0(5.0(0.0(3.0(x1))))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 0^1.1(1.0(2.1(4.0(5.0(0.0(3.1(x1))))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 1^1.0(2.1(4.0(5.0(0.0(3.0(x1)))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 1^1.0(2.1(4.0(5.0(0.0(3.1(x1)))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 2^1.1(4.0(5.0(0.0(3.0(x1))))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 2^1.1(4.0(5.0(0.0(3.1(x1))))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 4^1.0(5.0(0.0(3.0(x1)))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 4^1.0(5.0(0.0(3.1(x1)))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 5^1.0(0.0(3.0(x1))) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 5^1.0(0.0(3.1(x1))) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 0^1.0(3.0(x1)) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 0^1.0(3.1(x1)) 5^1.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3^1.0(x1) 5^1.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3^1.1(x1) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2^1.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2^1.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2^1.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1))))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2^1.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1))))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 0^1.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 0^1.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 3^1.0(2.0(3.1(4.0(3.1(4.1(4.0(x1))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 3^1.0(2.0(3.1(4.0(3.1(4.1(4.1(x1))))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2^1.0(3.1(4.0(3.1(4.1(4.0(x1)))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2^1.0(3.1(4.0(3.1(4.1(4.1(x1)))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 3^1.1(4.0(3.1(4.1(4.0(x1))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 3^1.1(4.0(3.1(4.1(4.1(x1))))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 4^1.0(3.1(4.1(4.0(x1)))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.0(3.1(4.1(4.1(x1)))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 3^1.1(4.1(4.0(x1))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 3^1.1(4.1(4.1(x1))) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 4^1.1(4.0(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(4.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 4^1.0(x1) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(x1) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2^1.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2^1.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 3^1.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1))))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 3^1.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1))))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 3^1.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 3^1.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 1^1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 1^1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1))))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 4^1.0(2.0(3.0(3.0(0.0(5.0(x1)))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 4^1.0(2.0(3.0(3.0(0.0(5.1(x1)))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2^1.0(3.0(3.0(0.0(5.0(x1))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2^1.0(3.0(3.0(0.0(5.1(x1))))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 3^1.0(3.0(0.0(5.0(x1)))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 3^1.0(3.0(0.0(5.1(x1)))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 3^1.0(0.0(5.0(x1))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 3^1.0(0.0(5.1(x1))) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 0^1.0(5.0(x1)) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 0^1.0(5.1(x1)) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 5^1.0(x1) 5^1.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 5^1.1(x1) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5^1.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5^1.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5^1.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1))))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5^1.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1))))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 0^1.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 0^1.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 0^1.0(2.1(4.0(5.0(2.0(5.0(2.0(x1))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 0^1.0(2.1(4.0(5.0(2.0(5.0(2.1(x1))))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 2^1.1(4.0(5.0(2.0(5.0(2.0(x1)))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 2^1.1(4.0(5.0(2.0(5.0(2.1(x1)))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 4^1.0(5.0(2.0(5.0(2.0(x1))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 4^1.0(5.0(2.0(5.0(2.1(x1))))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5^1.0(2.0(5.0(2.0(x1)))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5^1.0(2.0(5.0(2.1(x1)))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 2^1.0(5.0(2.0(x1))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 2^1.0(5.0(2.1(x1))) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5^1.0(2.0(x1)) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5^1.0(2.1(x1)) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 2^1.0(x1) 5^1.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 2^1.1(x1) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5^1.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5^1.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5^1.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1))))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5^1.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1))))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 3^1.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 3^1.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 1^1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 1^1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1))))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 3^1.0(5.0(5.0(3.0(3.1(4.0(x1)))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 3^1.0(5.0(5.0(3.0(3.1(4.1(x1)))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5^1.0(5.0(3.0(3.1(4.0(x1))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5^1.0(5.0(3.0(3.1(4.1(x1))))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5^1.0(3.0(3.1(4.0(x1)))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5^1.0(3.0(3.1(4.1(x1)))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 3^1.0(3.1(4.0(x1))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 3^1.0(3.1(4.1(x1))) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 3^1.1(4.0(x1)) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 3^1.1(4.1(x1)) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 4^1.0(x1) 5^1.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 4^1.1(x1) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3^1.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3^1.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 0^1.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1))))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 0^1.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1))))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3^1.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3^1.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3^1.1(4.0(3.1(1.0(2.0(2.0(3.0(x1))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3^1.1(4.0(3.1(1.0(2.0(2.0(3.1(x1))))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 4^1.0(3.1(1.0(2.0(2.0(3.0(x1)))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 4^1.0(3.1(1.0(2.0(2.0(3.1(x1)))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3^1.1(1.0(2.0(2.0(3.0(x1))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3^1.1(1.0(2.0(2.0(3.1(x1))))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 1^1.0(2.0(2.0(3.0(x1)))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 1^1.0(2.0(2.0(3.1(x1)))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 2^1.0(2.0(3.0(x1))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 2^1.0(2.0(3.1(x1))) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 2^1.0(3.0(x1)) 5^1.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 2^1.0(3.1(x1)) 0^1.0(2.1(1.0(x1))) -> 4^1.0(5.0(3.0(3.0(0.1(4.0(x1)))))) 0^1.0(2.1(1.1(x1))) -> 4^1.0(5.0(3.0(3.0(0.1(4.1(x1)))))) 0^1.0(2.1(1.0(x1))) -> 5^1.0(3.0(3.0(0.1(4.0(x1))))) 0^1.0(2.1(1.1(x1))) -> 5^1.0(3.0(3.0(0.1(4.1(x1))))) 0^1.0(2.1(1.0(x1))) -> 3^1.0(3.0(0.1(4.0(x1)))) 0^1.0(2.1(1.1(x1))) -> 3^1.0(3.0(0.1(4.1(x1)))) 0^1.0(2.1(1.0(x1))) -> 3^1.0(0.1(4.0(x1))) 0^1.0(2.1(1.1(x1))) -> 3^1.0(0.1(4.1(x1))) 0^1.0(2.1(1.0(x1))) -> 0^1.1(4.0(x1)) 0^1.0(2.1(1.1(x1))) -> 0^1.1(4.1(x1)) 0^1.0(2.1(1.0(x1))) -> 4^1.0(x1) 0^1.0(2.1(1.1(x1))) -> 4^1.1(x1) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 2^1.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 2^1.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 1^1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1))))))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 1^1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1))))))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 1^1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 1^1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.0(3.1(4.0(5.1(4.0(5.0(3.0(x1))))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 3^1.0(3.1(4.0(5.1(4.0(5.0(3.1(x1))))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.1(4.0(5.1(4.0(5.0(3.0(x1)))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 3^1.1(4.0(5.1(4.0(5.0(3.1(x1)))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 4^1.0(5.1(4.0(5.0(3.0(x1))))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 4^1.0(5.1(4.0(5.0(3.1(x1))))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 5^1.1(4.0(5.0(3.0(x1)))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 5^1.1(4.0(5.0(3.1(x1)))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 4^1.0(5.0(3.0(x1))) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 4^1.0(5.0(3.1(x1))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 5^1.0(3.0(x1)) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 5^1.0(3.1(x1)) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.0(x1) 0^1.1(1.0(0.1(1.0(2.1(x1))))) -> 3^1.1(x1) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2^1.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2^1.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 3^1.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1))))))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 3^1.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1))))))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 5^1.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 5^1.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2^1.0(2.0(3.0(2.0(0.1(4.1(4.0(x1))))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2^1.0(2.0(3.0(2.0(0.1(4.1(4.1(x1))))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2^1.0(3.0(2.0(0.1(4.1(4.0(x1)))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2^1.0(3.0(2.0(0.1(4.1(4.1(x1)))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 3^1.0(2.0(0.1(4.1(4.0(x1))))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 3^1.0(2.0(0.1(4.1(4.1(x1))))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2^1.0(0.1(4.1(4.0(x1)))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2^1.0(0.1(4.1(4.1(x1)))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 0^1.1(4.1(4.0(x1))) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 0^1.1(4.1(4.1(x1))) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 4^1.1(4.0(x1)) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(4.1(x1)) 0^1.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 4^1.0(x1) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(x1) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1))))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1))))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 2^1.0(5.0(3.0(5.0(5.1(1.0(0.0(x1))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 2^1.0(5.0(3.0(5.0(5.1(1.0(0.1(x1))))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.0(3.0(5.0(5.1(1.0(0.0(x1)))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.0(3.0(5.0(5.1(1.0(0.1(x1)))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 3^1.0(5.0(5.1(1.0(0.0(x1))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 3^1.0(5.0(5.1(1.0(0.1(x1))))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.0(5.1(1.0(0.0(x1)))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.0(5.1(1.0(0.1(x1)))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5^1.1(1.0(0.0(x1))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5^1.1(1.0(0.1(x1))) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 1^1.0(0.0(x1)) 0^1.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 1^1.0(0.1(x1)) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2^1.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2^1.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 5^1.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1))))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1))))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 4^1.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 4^1.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 5^1.1(4.0(3.1(4.1(4.0(5.1(1.0(x1))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(4.0(3.1(4.1(4.0(5.1(1.1(x1))))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 4^1.0(3.1(4.1(4.0(5.1(1.0(x1)))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 4^1.0(3.1(4.1(4.0(5.1(1.1(x1)))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 3^1.1(4.1(4.0(5.1(1.0(x1))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 3^1.1(4.1(4.0(5.1(1.1(x1))))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 4^1.1(4.0(5.1(1.0(x1)))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 4^1.1(4.0(5.1(1.1(x1)))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 4^1.0(5.1(1.0(x1))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 4^1.0(5.1(1.1(x1))) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 5^1.1(1.0(x1)) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(1.1(x1)) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3^1.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3^1.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 5^1.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1))))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 5^1.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1))))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 4^1.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 4^1.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 1^1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 1^1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1))))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 4^1.0(3.0(5.1(1.0(3.1(1.0(x1)))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 4^1.0(3.0(5.1(1.0(3.1(1.1(x1)))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3^1.0(5.1(1.0(3.1(1.0(x1))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3^1.0(5.1(1.0(3.1(1.1(x1))))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 5^1.1(1.0(3.1(1.0(x1)))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 5^1.1(1.0(3.1(1.1(x1)))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 1^1.0(3.1(1.0(x1))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 1^1.0(3.1(1.1(x1))) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3^1.1(1.0(x1)) 0^1.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3^1.1(1.1(x1)) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 4^1.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 4^1.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.0(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.1(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.0(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.1(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 1^1.0(3.0(2.0(2.1(4.0(x1))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 1^1.0(3.0(2.0(2.1(4.1(x1))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.0(2.0(2.1(4.0(x1)))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.0(2.0(2.1(4.1(x1)))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.0(2.1(4.0(x1))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.0(2.1(4.1(x1))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.1(4.0(x1)) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.1(4.1(x1)) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 3^1.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 3^1.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 4^1.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1))))))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 4^1.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1))))))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 0^1.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 0^1.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 5^1.1(4.0(2.0(2.1(1.0(3.0(3.0(x1))))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 5^1.1(4.0(2.0(2.1(1.0(3.0(3.1(x1))))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 4^1.0(2.0(2.1(1.0(3.0(3.0(x1)))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 4^1.0(2.0(2.1(1.0(3.0(3.1(x1)))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 2^1.0(2.1(1.0(3.0(3.0(x1))))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 2^1.0(2.1(1.0(3.0(3.1(x1))))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 2^1.1(1.0(3.0(3.0(x1)))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 2^1.1(1.0(3.0(3.1(x1)))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 1^1.0(3.0(3.0(x1))) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 1^1.0(3.0(3.1(x1))) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 3^1.0(3.0(x1)) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 3^1.0(3.1(x1)) 2^1.0(0.0(3.0(0.0(2.0(x1))))) -> 3^1.0(x1) 2^1.0(0.0(3.0(0.0(2.1(x1))))) -> 3^1.1(x1) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 2^1.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 2^1.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 3^1.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1))))))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 3^1.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1))))))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 5^1.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 5^1.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 2^1.0(3.0(3.0(0.0(3.0(5.0(3.0(x1))))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 2^1.0(3.0(3.0(0.0(3.0(5.0(3.1(x1))))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 3^1.0(3.0(0.0(3.0(5.0(3.0(x1)))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 3^1.0(3.0(0.0(3.0(5.0(3.1(x1)))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 3^1.0(0.0(3.0(5.0(3.0(x1))))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 3^1.0(0.0(3.0(5.0(3.1(x1))))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 0^1.0(3.0(5.0(3.0(x1)))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 0^1.0(3.0(5.0(3.1(x1)))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 3^1.0(5.0(3.0(x1))) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 3^1.0(5.0(3.1(x1))) 2^1.0(0.0(2.1(4.0(3.0(x1))))) -> 5^1.0(3.0(x1)) 2^1.0(0.0(2.1(4.0(3.1(x1))))) -> 5^1.0(3.1(x1)) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5^1.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5^1.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5^1.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1))))))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5^1.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1))))))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 2^1.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 2^1.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 2^1.0(2.0(3.1(1.1(4.1(4.0(3.0(x1))))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 2^1.0(2.0(3.1(1.1(4.1(4.0(3.1(x1))))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 2^1.0(3.1(1.1(4.1(4.0(3.0(x1)))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 2^1.0(3.1(1.1(4.1(4.0(3.1(x1)))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 3^1.1(1.1(4.1(4.0(3.0(x1))))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 3^1.1(1.1(4.1(4.0(3.1(x1))))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 1^1.1(4.1(4.0(3.0(x1)))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 1^1.1(4.1(4.0(3.1(x1)))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 4^1.1(4.0(3.0(x1))) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 4^1.1(4.0(3.1(x1))) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 4^1.0(3.0(x1)) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 4^1.0(3.1(x1)) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 3^1.0(x1) 2^1.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 3^1.1(x1) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0^1.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0^1.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 4^1.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1))))))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 4^1.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1))))))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 4^1.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 4^1.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 4^1.0(3.0(2.0(3.1(1.0(5.1(1.0(x1))))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 4^1.0(3.0(2.0(3.1(1.0(5.1(1.1(x1))))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 3^1.0(2.0(3.1(1.0(5.1(1.0(x1)))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 3^1.0(2.0(3.1(1.0(5.1(1.1(x1)))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 2^1.0(3.1(1.0(5.1(1.0(x1))))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 2^1.0(3.1(1.0(5.1(1.1(x1))))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 3^1.1(1.0(5.1(1.0(x1)))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 3^1.1(1.0(5.1(1.1(x1)))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 1^1.0(5.1(1.0(x1))) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 1^1.0(5.1(1.1(x1))) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 5^1.1(1.0(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 5^1.1(1.1(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 1^1.0(x1) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2^1.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 3^1.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1))))))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 3^1.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1))))))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 3^1.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 3^1.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 0^1.0(5.0(3.0(5.0(2.1(4.0(2.0(x1))))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 0^1.0(5.0(3.0(5.0(2.1(4.0(2.1(x1))))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 5^1.0(3.0(5.0(2.1(4.0(2.0(x1)))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 5^1.0(3.0(5.0(2.1(4.0(2.1(x1)))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 3^1.0(5.0(2.1(4.0(2.0(x1))))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 3^1.0(5.0(2.1(4.0(2.1(x1))))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 5^1.0(2.1(4.0(2.0(x1)))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 5^1.0(2.1(4.0(2.1(x1)))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2^1.1(4.0(2.0(x1))) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2^1.1(4.0(2.1(x1))) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 4^1.0(2.0(x1)) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 4^1.0(2.1(x1)) 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(x1) 2^1.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2^1.1(x1) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3^1.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3^1.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 2^1.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1))))))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 2^1.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1))))))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3^1.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3^1.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 2^1.1(4.1(4.1(4.0(3.0(0.1(4.0(x1))))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 2^1.1(4.1(4.1(4.0(3.0(0.1(4.1(x1))))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 4^1.1(4.1(4.0(3.0(0.1(4.0(x1)))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 4^1.1(4.1(4.0(3.0(0.1(4.1(x1)))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 4^1.1(4.0(3.0(0.1(4.0(x1))))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 4^1.1(4.0(3.0(0.1(4.1(x1))))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 4^1.0(3.0(0.1(4.0(x1)))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 4^1.0(3.0(0.1(4.1(x1)))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3^1.0(0.1(4.0(x1))) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3^1.0(0.1(4.1(x1))) 2^1.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 0^1.1(4.0(x1)) 2^1.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 0^1.1(4.1(x1)) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2^1.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2^1.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 5^1.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1))))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 5^1.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1))))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 5^1.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 5^1.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 1^1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 1^1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1))))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 0^1.0(3.0(3.0(2.0(3.0(3.0(x1)))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 0^1.0(3.0(3.0(2.0(3.0(3.1(x1)))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(3.0(2.0(3.0(3.0(x1))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 3^1.0(3.0(2.0(3.0(3.1(x1))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(2.0(3.0(3.0(x1)))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 3^1.0(2.0(3.0(3.1(x1)))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2^1.0(3.0(3.0(x1))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2^1.0(3.0(3.1(x1))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(3.0(x1)) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 3^1.0(3.1(x1)) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(x1) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 3^1.1(x1) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 499 less nodes. ---------------------------------------- (11) Complex Obligation (AND) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 0^1.1(4.1(4.1(x1))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(x1) 0^1.0(2.1(1.0(x1))) -> 0^1.1(4.0(x1)) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(4.1(x1)) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(x1) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(x1) 0^1.0(2.1(1.1(x1))) -> 0^1.1(4.1(x1)) 0^1.0(2.1(1.1(x1))) -> 4^1.1(x1) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(1.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(4.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(x1) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.0(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.1(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.0(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.1(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.1(4.0(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 5^1.1(1.1(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.0(x1))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.1(x1))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.0(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.1(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 2^1.1(4.0(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 2^1.1(4.1(x1)) 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 3^1.0(x1) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 0^1.0(3.0(3.0(2.0(3.0(3.0(x1)))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(x1) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.1(x1) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 3^1.0(0.0(x1)) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.1(4.1(x1)) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 2^1.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 3^1.0(x1) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 0^1.0(3.0(3.0(2.0(3.0(3.0(x1)))))) 2^1.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 3^1.0(x1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 1 POL(0.1(x_1)) = 1 + x_1 POL(0^1.0(x_1)) = 0 POL(0^1.1(x_1)) = 0 POL(1.0(x_1)) = x_1 POL(1.1(x_1)) = 1 + x_1 POL(1^1.1(x_1)) = 0 POL(2.0(x_1)) = 0 POL(2.1(x_1)) = 1 POL(2^1.1(x_1)) = x_1 POL(3.0(x_1)) = 0 POL(3.1(x_1)) = 0 POL(3^1.0(x_1)) = 0 POL(4.0(x_1)) = 0 POL(4.1(x_1)) = 0 POL(4^1.1(x_1)) = 0 POL(5.0(x_1)) = 0 POL(5.1(x_1)) = 0 POL(5^1.1(x_1)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 0^1.1(4.1(4.1(x1))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(x1) 0^1.0(2.1(1.0(x1))) -> 0^1.1(4.0(x1)) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(4.1(x1)) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(x1) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(x1) 0^1.0(2.1(1.1(x1))) -> 0^1.1(4.1(x1)) 0^1.0(2.1(1.1(x1))) -> 4^1.1(x1) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(1.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(4.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(x1) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.0(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.1(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.0(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.1(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.1(4.0(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 5^1.1(1.1(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.0(x1))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.1(x1))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.0(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.1(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 2^1.1(4.0(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 2^1.1(4.1(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.1(x1) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 3^1.0(0.0(x1)) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.1(4.1(x1)) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 2^1.1(4.1(x1)) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2^1.1(4.1(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0.0(x_1)) = 0 POL(0.1(x_1)) = 0 POL(0^1.0(x_1)) = 1 POL(0^1.1(x_1)) = 1 POL(1.0(x_1)) = 1 POL(1.1(x_1)) = x_1 POL(1^1.1(x_1)) = 1 POL(2.0(x_1)) = 0 POL(2.1(x_1)) = 0 POL(2^1.1(x_1)) = x_1 POL(3.0(x_1)) = 0 POL(3.1(x_1)) = 0 POL(3^1.0(x_1)) = 1 POL(4.0(x_1)) = 1 POL(4.1(x_1)) = 0 POL(4^1.1(x_1)) = 1 POL(5.0(x_1)) = 0 POL(5.1(x_1)) = 0 POL(5^1.1(x_1)) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 0^1.1(4.1(4.1(x1))) 0^1.1(1.0(0.1(1.0(2.0(x1))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(x1) 0^1.0(2.1(1.0(x1))) -> 0^1.1(4.0(x1)) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(4.1(x1)) 4^1.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 3^1.0(x1) 3^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(x1) 0^1.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 4^1.1(x1) 0^1.0(2.1(1.1(x1))) -> 0^1.1(4.1(x1)) 0^1.0(2.1(1.1(x1))) -> 4^1.1(x1) 0^1.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 5^1.1(1.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(4.1(x1)) 5^1.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 4^1.1(x1) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 3^1.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.0(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.0(0.1(1.0(3.0(2.0(2.1(4.1(x1))))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.0(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0^1.1(1.0(3.0(2.0(2.1(4.1(x1)))))) 0^1.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2^1.1(4.0(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 5^1.1(1.1(x1)) 2^1.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.0(x1))))) 1^1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 5^1.1(1.1(4.0(3.0(2.1(x1))))) 1^1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 1^1.1(x1) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.0(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 0^1.1(1.1(4.0(5.0(2.1(4.1(x1)))))) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 2^1.1(4.0(x1)) 1^1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 4^1.1(x1) 1^1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 3^1.0(0.0(x1)) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) PisEmptyProof (SOUND) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (18) TRUE ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(x1) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: 2^1.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2^1.0(x1) The following rules are removed from R: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(0.1(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(2^1.0(x_1)) = x_1 POL(3.0(x_1)) = x_1 ---------------------------------------- (21) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 5^1.0(x1) The TRS R consists of the following rules: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: 5^1.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 5^1.0(x1) The following rules are removed from R: 0.0(2.1(1.0(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.0(x1)))))))))) 0.0(2.1(1.1(x1))) -> 3.1(4.1(1.0(5.1(4.0(5.0(3.0(3.0(0.1(4.1(x1)))))))))) 5.1(4.0(0.0(0.1(1.0(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.0(x1)))))))))) 5.1(4.0(0.0(0.1(1.1(x1))))) -> 3.0(2.0(5.1(4.1(1.0(3.1(1.0(3.1(4.1(1.1(x1)))))))))) 2.0(0.0(3.0(0.0(2.0(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.0(x1)))))))))) 2.0(0.0(3.0(0.0(2.1(x1))))) -> 3.1(4.0(0.0(5.1(4.0(2.0(2.1(1.0(3.0(3.1(x1)))))))))) 0.1(1.0(0.1(1.0(2.0(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.0(x1)))))))))) 0.1(1.0(0.1(1.0(2.1(x1))))) -> 2.1(1.1(1.0(3.0(3.1(4.0(5.1(4.0(5.0(3.1(x1)))))))))) 2.0(0.0(2.1(4.0(3.0(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.0(x1)))))))))) 2.0(0.0(2.1(4.0(3.1(x1))))) -> 2.0(3.0(5.0(2.0(3.0(3.0(0.0(3.0(5.0(3.1(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.0(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.0(x1)))))))))) 0.1(4.0(2.0(5.0(3.0(0.1(x1)))))) -> 2.0(3.0(5.0(2.0(2.0(3.0(2.0(0.1(4.1(4.1(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.0(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.0(x1)))))))))) 5.1(4.0(0.0(2.1(1.1(1.1(x1)))))) -> 3.0(3.0(2.0(0.1(1.0(2.1(4.0(5.0(0.0(3.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.0(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.0(2.1(x1)))))) -> 5.0(5.0(2.0(2.0(2.0(3.1(1.1(4.1(4.0(3.1(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.0(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.0(x1)))))))))) 1.1(1.1(1.0(0.0(2.0(2.1(x1)))))) -> 4.0(5.0(2.0(2.0(2.0(5.1(1.1(4.0(3.0(2.1(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.0(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.0(x1)))))))))) 2.1(4.0(0.1(1.1(4.0(2.1(x1)))))) -> 0.1(4.1(4.1(4.0(3.0(2.0(3.1(1.0(5.1(1.1(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.0(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.0(x1)))))))))) 1.1(1.1(1.0(2.1(4.0(2.1(x1)))))) -> 4.1(4.1(1.0(3.1(4.0(3.1(4.0(5.0(3.1(1.1(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.0(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.0(x1)))))))))) 2.0(0.0(0.1(1.0(0.0(3.1(x1)))))) -> 2.0(3.0(3.0(0.0(5.0(3.0(5.0(2.1(4.0(2.1(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.0(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.0(x1)))))))))) 1.1(1.1(1.1(1.0(0.0(3.1(x1)))))) -> 1.0(5.1(4.0(2.0(5.1(4.1(4.0(2.0(2.0(3.1(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.0(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.0(x1)))))))))) 2.0(0.0(2.1(1.1(1.1(4.1(x1)))))) -> 3.0(2.0(3.0(2.1(4.1(4.1(4.0(3.0(0.1(4.1(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.0(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.0(x1)))))))))) 0.1(1.1(1.1(1.1(1.0(2.0(0.1(x1))))))) -> 5.0(5.0(5.0(2.0(5.0(3.0(5.0(5.1(1.0(0.1(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.0(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.0(x1)))))))))) 5.1(1.1(1.1(1.1(4.0(2.0(0.1(x1))))))) -> 2.0(2.0(0.0(3.0(2.0(3.1(4.0(3.1(4.1(4.1(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.0(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.0(x1)))))))))) 1.1(4.0(0.0(0.0(2.1(4.0(0.1(x1))))))) -> 1.0(3.0(3.1(4.0(0.1(1.1(4.0(5.0(2.1(4.1(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.0(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.0(x1)))))))))) 1.1(4.0(5.0(0.0(3.1(4.0(0.1(x1))))))) -> 1.1(4.0(5.0(2.1(1.0(3.0(5.1(1.0(3.0(0.1(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.0(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.0(x1)))))))))) 0.0(0.0(2.0(2.0(5.0(0.1(1.1(x1))))))) -> 2.0(5.1(4.0(5.1(4.0(3.1(4.1(4.0(5.1(1.1(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.0(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.0(x1)))))))))) 0.0(0.0(5.1(4.0(3.1(1.1(1.1(x1))))))) -> 3.0(5.1(4.1(1.1(4.0(3.0(5.1(1.0(3.1(1.1(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.0(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.0(x1)))))))))) 5.0(0.0(5.1(1.0(3.1(4.1(1.1(x1))))))) -> 2.0(3.0(3.1(1.1(4.0(2.0(3.0(3.0(0.0(5.1(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.0(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.0(x1)))))))))) 5.0(5.0(0.0(2.0(0.0(5.1(1.1(x1))))))) -> 5.0(5.0(0.0(0.0(2.1(4.0(5.0(2.0(5.0(2.1(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.0(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.0(x1)))))))))) 5.0(0.0(2.0(5.1(1.0(0.0(2.1(x1))))))) -> 5.0(5.0(3.1(1.0(3.0(5.0(5.0(3.0(3.1(4.1(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.0(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.0(x1)))))))))) 4.1(1.1(4.0(5.0(0.1(4.0(2.1(x1))))))) -> 4.1(1.0(5.0(3.0(5.1(1.0(5.0(5.1(4.0(3.1(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.0(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.0(x1)))))))))) 5.1(4.0(0.1(4.1(1.1(4.0(3.1(x1))))))) -> 3.0(0.0(3.0(3.1(4.0(3.1(1.0(2.0(2.0(3.1(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.0(x1)))))))))) 0.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 0.1(4.0(3.0(0.0(0.1(1.0(3.0(2.0(2.1(4.1(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.0(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.0(x1)))))))))) 3.0(0.0(2.1(4.0(0.1(1.1(4.1(x1))))))) -> 2.0(3.1(4.0(5.1(1.0(0.0(0.0(3.1(4.0(0.1(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.0(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.0(x1)))))))))) 2.1(1.0(0.1(1.1(1.1(1.1(4.1(x1))))))) -> 2.0(5.0(5.1(1.0(0.0(3.0(3.0(2.0(3.0(3.1(x1)))))))))) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0.0(x_1)) = x_1 POL(1.0(x_1)) = x_1 POL(3.1(x_1)) = x_1 POL(4.1(x_1)) = x_1 POL(5.1(x_1)) = x_1 POL(5^1.0(x_1)) = x_1 ---------------------------------------- (26) Obligation: Q DP problem: P is empty. R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (28) YES ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) 0^1(1(0(1(2(x1))))) -> 3^1(x1) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(x1) 0^1(2(1(x1))) -> 0^1(4(x1)) 0^1(4(2(5(3(0(x1)))))) -> 4^1(4(x1)) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(x1) 0^1(4(2(5(3(0(x1)))))) -> 4^1(x1) 0^1(2(1(x1))) -> 4^1(x1) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(1(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(4(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(x1) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(4(3(0(0(1(3(2(2(4(x1)))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(0(0(1(3(2(2(4(x1)))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(1(3(2(2(4(x1))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(1(3(2(2(4(x1)))))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(4(x1)) 2^1(4(0(1(4(2(x1)))))) -> 5^1(1(x1)) 2^1(4(0(1(4(2(x1)))))) -> 1^1(x1) 1^1(1(1(0(2(2(x1)))))) -> 5^1(1(4(3(2(x1))))) 1^1(1(1(2(4(2(x1)))))) -> 1^1(x1) 1^1(4(0(0(2(4(0(x1))))))) -> 0^1(1(4(5(2(4(x1)))))) 1^1(4(0(0(2(4(0(x1))))))) -> 2^1(4(x1)) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(x1) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(0(x1)) The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(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. 0^1(1(0(1(2(x1))))) -> 3^1(x1) 3^1(0(2(4(0(1(4(x1))))))) -> 0^1(x1) 0^1(2(1(x1))) -> 0^1(4(x1)) 0^1(4(2(5(3(0(x1)))))) -> 4^1(4(x1)) 4^1(1(4(5(0(4(2(x1))))))) -> 3^1(x1) 0^1(4(2(5(3(0(x1)))))) -> 4^1(x1) 0^1(2(1(x1))) -> 4^1(x1) 0^1(0(2(2(5(0(1(x1))))))) -> 5^1(1(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(4(x1)) 5^1(1(1(1(4(2(0(x1))))))) -> 4^1(x1) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(4(3(0(0(1(3(2(2(4(x1)))))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 3^1(0(0(1(3(2(2(4(x1)))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(0(1(3(2(2(4(x1))))))) 0^1(0(2(4(0(1(4(x1))))))) -> 0^1(1(3(2(2(4(x1)))))) 0^1(0(2(4(0(1(4(x1))))))) -> 2^1(4(x1)) 2^1(4(0(1(4(2(x1)))))) -> 1^1(x1) 1^1(1(1(0(2(2(x1)))))) -> 5^1(1(4(3(2(x1))))) 1^1(1(1(2(4(2(x1)))))) -> 1^1(x1) 1^1(4(0(0(2(4(0(x1))))))) -> 0^1(1(4(5(2(4(x1)))))) 1^1(4(0(0(2(4(0(x1))))))) -> 2^1(4(x1)) 1^1(4(0(0(2(4(0(x1))))))) -> 4^1(x1) 1^1(4(5(0(3(4(0(x1))))))) -> 3^1(0(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = x_1 + 2 POL( 2^1_1(x_1) ) = max{0, x_1 - 2} POL( 4^1_1(x_1) ) = max{0, x_1 - 1} POL( 5^1_1(x_1) ) = x_1 POL( 0_1(x_1) ) = x_1 + 2 POL( 3^1_1(x_1) ) = max{0, x_1 - 2} POL( 2_1(x_1) ) = x_1 POL( 3_1(x_1) ) = max{0, x_1 - 2} POL( 4_1(x_1) ) = x_1 POL( 1_1(x_1) ) = x_1 + 2 POL( 5_1(x_1) ) = x_1 POL( 1^1_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) 2^1(4(0(1(4(2(x1)))))) -> 5^1(1(x1)) The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. 0^1(4(2(5(3(0(x1)))))) -> 0^1(4(4(x1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( 0^1_1(x_1) ) = x_1 + 1 POL( 4_1(x_1) ) = x_1 POL( 1_1(x_1) ) = x_1 + 1 POL( 5_1(x_1) ) = max{0, x_1 - 1} POL( 0_1(x_1) ) = x_1 + 2 POL( 2_1(x_1) ) = x_1 + 1 POL( 3_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) ---------------------------------------- (35) Obligation: Q DP problem: P is empty. The TRS R consists of the following rules: 0(2(1(x1))) -> 3(4(1(5(4(5(3(3(0(4(x1)))))))))) 5(4(0(0(1(x1))))) -> 3(2(5(4(1(3(1(3(4(1(x1)))))))))) 2(0(3(0(2(x1))))) -> 3(4(0(5(4(2(2(1(3(3(x1)))))))))) 0(1(0(1(2(x1))))) -> 2(1(1(3(3(4(5(4(5(3(x1)))))))))) 2(0(2(4(3(x1))))) -> 2(3(5(2(3(3(0(3(5(3(x1)))))))))) 0(4(2(5(3(0(x1)))))) -> 2(3(5(2(2(3(2(0(4(4(x1)))))))))) 5(4(0(2(1(1(x1)))))) -> 3(3(2(0(1(2(4(5(0(3(x1)))))))))) 2(1(0(1(1(2(x1)))))) -> 5(5(2(2(2(3(1(4(4(3(x1)))))))))) 1(1(1(0(2(2(x1)))))) -> 4(5(2(2(2(5(1(4(3(2(x1)))))))))) 2(4(0(1(4(2(x1)))))) -> 0(4(4(4(3(2(3(1(5(1(x1)))))))))) 1(1(1(2(4(2(x1)))))) -> 4(4(1(3(4(3(4(5(3(1(x1)))))))))) 2(0(0(1(0(3(x1)))))) -> 2(3(3(0(5(3(5(2(4(2(x1)))))))))) 1(1(1(1(0(3(x1)))))) -> 1(5(4(2(5(4(4(2(2(3(x1)))))))))) 2(0(2(1(1(4(x1)))))) -> 3(2(3(2(4(4(4(3(0(4(x1)))))))))) 0(1(1(1(1(2(0(x1))))))) -> 5(5(5(2(5(3(5(5(1(0(x1)))))))))) 5(1(1(1(4(2(0(x1))))))) -> 2(2(0(3(2(3(4(3(4(4(x1)))))))))) 1(4(0(0(2(4(0(x1))))))) -> 1(3(3(4(0(1(4(5(2(4(x1)))))))))) 1(4(5(0(3(4(0(x1))))))) -> 1(4(5(2(1(3(5(1(3(0(x1)))))))))) 0(0(2(2(5(0(1(x1))))))) -> 2(5(4(5(4(3(4(4(5(1(x1)))))))))) 0(0(5(4(3(1(1(x1))))))) -> 3(5(4(1(4(3(5(1(3(1(x1)))))))))) 5(0(5(1(3(4(1(x1))))))) -> 2(3(3(1(4(2(3(3(0(5(x1)))))))))) 5(5(0(2(0(5(1(x1))))))) -> 5(5(0(0(2(4(5(2(5(2(x1)))))))))) 5(0(2(5(1(0(2(x1))))))) -> 5(5(3(1(3(5(5(3(3(4(x1)))))))))) 4(1(4(5(0(4(2(x1))))))) -> 4(1(5(3(5(1(5(5(4(3(x1)))))))))) 5(4(0(4(1(4(3(x1))))))) -> 3(0(3(3(4(3(1(2(2(3(x1)))))))))) 0(0(2(4(0(1(4(x1))))))) -> 0(4(3(0(0(1(3(2(2(4(x1)))))))))) 3(0(2(4(0(1(4(x1))))))) -> 2(3(4(5(1(0(0(3(4(0(x1)))))))))) 2(1(0(1(1(1(4(x1))))))) -> 2(5(5(1(0(3(3(2(3(3(x1)))))))))) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (37) YES