42.42/11.84 YES 42.99/11.89 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.99/11.89 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.99/11.89 42.99/11.89 42.99/11.89 Termination w.r.t. Q of the given QTRS could be proven: 42.99/11.89 42.99/11.89 (0) QTRS 42.99/11.89 (1) QTRS Reverse [EQUIVALENT, 0 ms] 42.99/11.89 (2) QTRS 42.99/11.89 (3) DependencyPairsProof [EQUIVALENT, 247 ms] 42.99/11.89 (4) QDP 42.99/11.89 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 42.99/11.89 (6) QDP 42.99/11.89 (7) QDPOrderProof [EQUIVALENT, 334 ms] 42.99/11.89 (8) QDP 42.99/11.89 (9) QDPOrderProof [EQUIVALENT, 82 ms] 42.99/11.89 (10) QDP 42.99/11.89 (11) DependencyGraphProof [EQUIVALENT, 0 ms] 42.99/11.89 (12) QDP 42.99/11.89 (13) UsableRulesProof [EQUIVALENT, 0 ms] 42.99/11.89 (14) QDP 42.99/11.89 (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.99/11.89 (16) YES 42.99/11.89 42.99/11.89 42.99/11.89 ---------------------------------------- 42.99/11.89 42.99/11.89 (0) 42.99/11.89 Obligation: 42.99/11.89 Q restricted rewrite system: 42.99/11.89 The TRS R consists of the following rules: 42.99/11.89 42.99/11.89 0(1(1(2(x1)))) -> 0(1(0(3(1(2(x1)))))) 42.99/11.89 0(2(3(1(x1)))) -> 0(3(2(0(1(x1))))) 42.99/11.89 0(2(3(1(x1)))) -> 1(0(3(4(2(x1))))) 42.99/11.89 0(2(3(1(x1)))) -> 4(0(3(2(1(x1))))) 42.99/11.89 0(2(3(1(x1)))) -> 0(0(3(2(1(4(x1)))))) 42.99/11.89 3(0(1(1(x1)))) -> 0(3(4(1(1(0(x1)))))) 42.99/11.89 3(0(1(2(x1)))) -> 0(3(4(1(2(4(x1)))))) 42.99/11.89 3(0(1(2(x1)))) -> 0(3(4(4(1(2(x1)))))) 42.99/11.89 3(0(2(1(x1)))) -> 0(3(2(1(0(x1))))) 42.99/11.89 3(0(2(1(x1)))) -> 0(3(2(1(4(x1))))) 42.99/11.89 3(0(2(1(x1)))) -> 0(3(4(2(1(0(x1)))))) 42.99/11.89 0(1(0(1(2(x1))))) -> 0(0(2(1(1(3(x1)))))) 42.99/11.89 0(1(1(4(2(x1))))) -> 1(0(3(1(4(2(x1)))))) 42.99/11.89 0(1(2(3(1(x1))))) -> 1(0(3(1(2(2(x1)))))) 42.99/11.89 0(1(2(3(1(x1))))) -> 1(4(0(3(1(2(x1)))))) 42.99/11.89 0(1(2(3(1(x1))))) -> 2(1(0(3(2(1(x1)))))) 42.99/11.89 0(1(5(3(1(x1))))) -> 0(3(1(5(4(1(x1)))))) 42.99/11.89 0(2(3(2(1(x1))))) -> 0(3(2(4(1(2(x1)))))) 42.99/11.89 0(2(4(3(1(x1))))) -> 0(3(4(1(2(0(x1)))))) 42.99/11.89 0(2(4(3(1(x1))))) -> 1(0(0(3(4(2(x1)))))) 42.99/11.89 0(3(3(1(2(x1))))) -> 3(0(1(0(3(2(x1)))))) 42.99/11.89 0(3(3(2(1(x1))))) -> 0(3(2(3(1(4(x1)))))) 42.99/11.89 0(5(2(3(1(x1))))) -> 0(3(2(1(5(4(x1)))))) 42.99/11.89 0(5(2(3(1(x1))))) -> 0(5(0(3(2(1(x1)))))) 42.99/11.89 0(5(4(3(1(x1))))) -> 0(3(4(1(4(5(x1)))))) 42.99/11.89 1(4(0(1(2(x1))))) -> 2(1(4(0(3(1(x1)))))) 42.99/11.89 3(0(1(0(2(x1))))) -> 0(3(2(0(4(1(x1)))))) 42.99/11.89 3(0(2(3(1(x1))))) -> 3(0(0(3(1(2(x1)))))) 42.99/11.89 3(0(2(5(1(x1))))) -> 0(3(2(5(0(1(x1)))))) 42.99/11.89 3(0(2(5(1(x1))))) -> 0(3(5(2(1(4(x1)))))) 42.99/11.89 3(0(2(5(1(x1))))) -> 5(4(0(3(2(1(x1)))))) 42.99/11.89 3(0(4(5(1(x1))))) -> 5(0(0(3(4(1(x1)))))) 42.99/11.89 3(3(0(1(4(x1))))) -> 3(0(0(3(1(4(x1)))))) 42.99/11.89 3(3(1(1(1(x1))))) -> 0(3(1(3(1(1(x1)))))) 42.99/11.89 3(3(1(1(2(x1))))) -> 1(3(2(1(4(3(x1)))))) 42.99/11.89 3(3(1(1(4(x1))))) -> 4(4(3(1(3(1(x1)))))) 42.99/11.89 3(4(3(2(1(x1))))) -> 3(0(3(2(4(1(x1)))))) 42.99/11.89 3(5(0(2(1(x1))))) -> 0(3(2(0(5(1(x1)))))) 42.99/11.89 3(5(0(2(1(x1))))) -> 0(3(5(2(4(1(x1)))))) 42.99/11.89 4(0(1(1(4(x1))))) -> 1(0(3(4(4(1(x1)))))) 42.99/11.89 4(5(3(2(1(x1))))) -> 0(3(1(5(2(4(x1)))))) 42.99/11.89 5(0(1(1(4(x1))))) -> 0(3(1(5(1(4(x1)))))) 42.99/11.89 5(0(2(3(1(x1))))) -> 5(0(3(2(1(0(x1)))))) 42.99/11.89 5(3(0(1(1(x1))))) -> 0(3(1(1(5(0(x1)))))) 42.99/11.89 5(3(0(2(1(x1))))) -> 0(3(4(2(1(5(x1)))))) 42.99/11.89 5(3(0(2(1(x1))))) -> 0(3(5(1(2(4(x1)))))) 42.99/11.89 5(3(1(1(2(x1))))) -> 3(1(2(1(4(5(x1)))))) 42.99/11.89 5(3(1(1(4(x1))))) -> 3(4(1(5(2(1(x1)))))) 42.99/11.89 5(4(3(4(1(x1))))) -> 0(3(4(4(1(5(x1)))))) 42.99/11.89 42.99/11.89 Q is empty. 42.99/11.89 42.99/11.89 ---------------------------------------- 42.99/11.89 42.99/11.89 (1) QTRS Reverse (EQUIVALENT) 42.99/11.89 We applied the QTRS Reverse Processor [REVERSE]. 42.99/11.89 ---------------------------------------- 42.99/11.89 42.99/11.89 (2) 42.99/11.89 Obligation: 42.99/11.89 Q restricted rewrite system: 42.99/11.89 The TRS R consists of the following rules: 42.99/11.89 42.99/11.89 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.89 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.89 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.89 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.89 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.89 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.89 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.89 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.89 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.89 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.89 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.89 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.89 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.89 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.89 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.89 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.89 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.89 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.89 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.89 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.89 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.89 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.89 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.89 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.89 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.89 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.89 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.89 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.89 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.89 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.89 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.89 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.89 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.89 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.89 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.89 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.89 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.89 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.89 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.89 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.89 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.89 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.89 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.89 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.89 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.89 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.89 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.89 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.89 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.89 42.99/11.89 Q is empty. 42.99/11.89 42.99/11.89 ---------------------------------------- 42.99/11.89 42.99/11.89 (3) DependencyPairsProof (EQUIVALENT) 42.99/11.89 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.99/11.89 ---------------------------------------- 42.99/11.89 42.99/11.89 (4) 42.99/11.89 Obligation: 42.99/11.89 Q DP problem: 42.99/11.89 The TRS P consists of the following rules: 42.99/11.89 42.99/11.89 2^1(1(1(0(x1)))) -> 2^1(1(3(0(1(0(x1)))))) 42.99/11.89 2^1(1(1(0(x1)))) -> 1^1(3(0(1(0(x1))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 1^1(0(2(3(0(x1))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 2^1(3(0(x1))) 42.99/11.89 1^1(3(2(0(x1)))) -> 2^1(4(3(0(1(x1))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 4^1(3(0(1(x1)))) 42.99/11.89 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.89 1^1(3(2(0(x1)))) -> 1^1(2(3(0(4(x1))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 2^1(3(0(4(x1)))) 42.99/11.89 1^1(3(2(0(x1)))) -> 4^1(x1) 42.99/11.89 1^1(3(2(0(x1)))) -> 4^1(1(2(3(0(0(x1)))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 1^1(2(3(0(0(x1))))) 42.99/11.89 1^1(3(2(0(x1)))) -> 2^1(3(0(0(x1)))) 42.99/11.89 1^1(1(0(3(x1)))) -> 1^1(1(4(3(0(x1))))) 42.99/11.89 1^1(1(0(3(x1)))) -> 1^1(4(3(0(x1)))) 42.99/11.89 1^1(1(0(3(x1)))) -> 4^1(3(0(x1))) 42.99/11.89 2^1(1(0(3(x1)))) -> 4^1(2(1(4(3(0(x1)))))) 42.99/11.89 2^1(1(0(3(x1)))) -> 2^1(1(4(3(0(x1))))) 42.99/11.89 2^1(1(0(3(x1)))) -> 1^1(4(3(0(x1)))) 42.99/11.89 2^1(1(0(3(x1)))) -> 4^1(3(0(x1))) 42.99/11.89 2^1(1(0(3(x1)))) -> 2^1(1(4(4(3(0(x1)))))) 42.99/11.89 2^1(1(0(3(x1)))) -> 1^1(4(4(3(0(x1))))) 42.99/11.89 2^1(1(0(3(x1)))) -> 4^1(4(3(0(x1)))) 42.99/11.89 1^1(2(0(3(x1)))) -> 1^1(2(3(0(x1)))) 42.99/11.89 1^1(2(0(3(x1)))) -> 2^1(3(0(x1))) 42.99/11.89 1^1(2(0(3(x1)))) -> 4^1(1(2(3(0(x1))))) 42.99/11.89 1^1(2(0(3(x1)))) -> 1^1(2(4(3(0(x1))))) 42.99/11.89 1^1(2(0(3(x1)))) -> 2^1(4(3(0(x1)))) 42.99/11.89 1^1(2(0(3(x1)))) -> 4^1(3(0(x1))) 42.99/11.89 2^1(1(0(1(0(x1))))) -> 1^1(1(2(0(0(x1))))) 42.99/11.89 2^1(1(0(1(0(x1))))) -> 1^1(2(0(0(x1)))) 42.99/11.89 2^1(1(0(1(0(x1))))) -> 2^1(0(0(x1))) 42.99/11.89 2^1(4(1(1(0(x1))))) -> 2^1(4(1(3(0(1(x1)))))) 42.99/11.89 2^1(4(1(1(0(x1))))) -> 4^1(1(3(0(1(x1))))) 42.99/11.89 2^1(4(1(1(0(x1))))) -> 1^1(3(0(1(x1)))) 42.99/11.89 2^1(4(1(1(0(x1))))) -> 1^1(x1) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 2^1(2(1(3(0(1(x1)))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 2^1(1(3(0(1(x1))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 1^1(3(0(1(x1)))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 1^1(x1) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 2^1(1(3(0(4(1(x1)))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 1^1(3(0(4(1(x1))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 4^1(1(x1)) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 1^1(2(3(0(1(2(x1)))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 2^1(3(0(1(2(x1))))) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 1^1(2(x1)) 42.99/11.89 1^1(3(2(1(0(x1))))) -> 2^1(x1) 42.99/11.89 1^1(3(5(1(0(x1))))) -> 1^1(4(5(1(3(0(x1)))))) 42.99/11.89 1^1(3(5(1(0(x1))))) -> 4^1(5(1(3(0(x1))))) 42.99/11.89 1^1(3(5(1(0(x1))))) -> 1^1(3(0(x1))) 42.99/11.89 1^1(2(3(2(0(x1))))) -> 2^1(1(4(2(3(0(x1)))))) 42.99/11.89 1^1(2(3(2(0(x1))))) -> 1^1(4(2(3(0(x1))))) 42.99/11.89 1^1(2(3(2(0(x1))))) -> 4^1(2(3(0(x1)))) 42.99/11.89 1^1(2(3(2(0(x1))))) -> 2^1(3(0(x1))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 2^1(1(4(3(0(x1))))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 1^1(4(3(0(x1)))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 4^1(3(0(x1))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 2^1(4(3(0(0(1(x1)))))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 4^1(3(0(0(1(x1))))) 42.99/11.89 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.89 2^1(1(3(3(0(x1))))) -> 2^1(3(0(1(0(3(x1)))))) 42.99/11.90 2^1(1(3(3(0(x1))))) -> 1^1(0(3(x1))) 42.99/11.90 1^1(2(3(3(0(x1))))) -> 4^1(1(3(2(3(0(x1)))))) 42.99/11.90 1^1(2(3(3(0(x1))))) -> 1^1(3(2(3(0(x1))))) 42.99/11.90 1^1(2(3(3(0(x1))))) -> 2^1(3(0(x1))) 42.99/11.90 1^1(3(2(5(0(x1))))) -> 4^1(5(1(2(3(0(x1)))))) 42.99/11.90 1^1(3(2(5(0(x1))))) -> 1^1(2(3(0(x1)))) 42.99/11.90 1^1(3(2(5(0(x1))))) -> 2^1(3(0(x1))) 42.99/11.90 1^1(3(2(5(0(x1))))) -> 1^1(2(3(0(5(0(x1)))))) 42.99/11.90 1^1(3(2(5(0(x1))))) -> 2^1(3(0(5(0(x1))))) 42.99/11.90 1^1(3(4(5(0(x1))))) -> 4^1(1(4(3(0(x1))))) 42.99/11.90 1^1(3(4(5(0(x1))))) -> 1^1(4(3(0(x1)))) 42.99/11.90 1^1(3(4(5(0(x1))))) -> 4^1(3(0(x1))) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 1^1(3(0(4(1(2(x1)))))) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 4^1(1(2(x1))) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 1^1(2(x1)) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 2^1(x1) 42.99/11.90 2^1(0(1(0(3(x1))))) -> 1^1(4(0(2(3(0(x1)))))) 42.99/11.90 2^1(0(1(0(3(x1))))) -> 4^1(0(2(3(0(x1))))) 42.99/11.90 2^1(0(1(0(3(x1))))) -> 2^1(3(0(x1))) 42.99/11.90 1^1(3(2(0(3(x1))))) -> 2^1(1(3(0(0(3(x1)))))) 42.99/11.90 1^1(3(2(0(3(x1))))) -> 1^1(3(0(0(3(x1))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 1^1(0(5(2(3(0(x1)))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 2^1(3(0(x1))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 4^1(1(2(5(3(0(x1)))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 1^1(2(5(3(0(x1))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 2^1(5(3(0(x1)))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 1^1(2(3(0(4(5(x1)))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 2^1(3(0(4(5(x1))))) 42.99/11.90 1^1(5(2(0(3(x1))))) -> 4^1(5(x1)) 42.99/11.90 1^1(5(4(0(3(x1))))) -> 1^1(4(3(0(0(5(x1)))))) 42.99/11.90 1^1(5(4(0(3(x1))))) -> 4^1(3(0(0(5(x1))))) 42.99/11.90 4^1(1(0(3(3(x1))))) -> 4^1(1(3(0(0(3(x1)))))) 42.99/11.90 4^1(1(0(3(3(x1))))) -> 1^1(3(0(0(3(x1))))) 42.99/11.90 1^1(1(1(3(3(x1))))) -> 1^1(1(3(1(3(0(x1)))))) 42.99/11.90 1^1(1(1(3(3(x1))))) -> 1^1(3(1(3(0(x1))))) 42.99/11.90 1^1(1(1(3(3(x1))))) -> 1^1(3(0(x1))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 4^1(1(2(3(1(x1))))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 1^1(2(3(1(x1)))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 2^1(3(1(x1))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 1^1(x1) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(1(3(4(4(x1)))))) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(4(4(x1)))) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(4(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(x1) 42.99/11.90 1^1(2(3(4(3(x1))))) -> 1^1(4(2(3(0(3(x1)))))) 42.99/11.90 1^1(2(3(4(3(x1))))) -> 4^1(2(3(0(3(x1))))) 42.99/11.90 1^1(2(3(4(3(x1))))) -> 2^1(3(0(3(x1)))) 42.99/11.90 1^1(2(0(5(3(x1))))) -> 1^1(5(0(2(3(0(x1)))))) 42.99/11.90 1^1(2(0(5(3(x1))))) -> 2^1(3(0(x1))) 42.99/11.90 1^1(2(0(5(3(x1))))) -> 1^1(4(2(5(3(0(x1)))))) 42.99/11.90 1^1(2(0(5(3(x1))))) -> 4^1(2(5(3(0(x1))))) 42.99/11.90 1^1(2(0(5(3(x1))))) -> 2^1(5(3(0(x1)))) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 1^1(4(4(3(0(1(x1)))))) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 4^1(4(3(0(1(x1))))) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 4^1(3(0(1(x1)))) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 1^1(x1) 42.99/11.90 1^1(2(3(5(4(x1))))) -> 4^1(2(5(1(3(0(x1)))))) 42.99/11.90 1^1(2(3(5(4(x1))))) -> 2^1(5(1(3(0(x1))))) 42.99/11.90 1^1(2(3(5(4(x1))))) -> 1^1(3(0(x1))) 42.99/11.90 4^1(1(1(0(5(x1))))) -> 4^1(1(5(1(3(0(x1)))))) 42.99/11.90 4^1(1(1(0(5(x1))))) -> 1^1(5(1(3(0(x1))))) 42.99/11.90 4^1(1(1(0(5(x1))))) -> 1^1(3(0(x1))) 42.99/11.90 1^1(3(2(0(5(x1))))) -> 1^1(2(3(0(5(x1))))) 42.99/11.90 1^1(3(2(0(5(x1))))) -> 2^1(3(0(5(x1)))) 42.99/11.90 1^1(1(0(3(5(x1))))) -> 1^1(1(3(0(x1)))) 42.99/11.90 1^1(1(0(3(5(x1))))) -> 1^1(3(0(x1))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 1^1(2(4(3(0(x1))))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 2^1(4(3(0(x1)))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 4^1(3(0(x1))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 4^1(2(1(5(3(0(x1)))))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 2^1(1(5(3(0(x1))))) 42.99/11.90 1^1(2(0(3(5(x1))))) -> 1^1(5(3(0(x1)))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 4^1(1(2(1(3(x1))))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(2(1(3(x1)))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 2^1(1(3(x1))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(3(x1)) 42.99/11.90 4^1(1(1(3(5(x1))))) -> 1^1(2(5(1(4(3(x1)))))) 42.99/11.90 4^1(1(1(3(5(x1))))) -> 2^1(5(1(4(3(x1))))) 42.99/11.90 4^1(1(1(3(5(x1))))) -> 1^1(4(3(x1))) 42.99/11.90 4^1(1(1(3(5(x1))))) -> 4^1(3(x1)) 42.99/11.90 1^1(4(3(4(5(x1))))) -> 1^1(4(4(3(0(x1))))) 42.99/11.90 1^1(4(3(4(5(x1))))) -> 4^1(4(3(0(x1)))) 42.99/11.90 1^1(4(3(4(5(x1))))) -> 4^1(3(0(x1))) 42.99/11.90 42.99/11.90 The TRS R consists of the following rules: 42.99/11.90 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (5) DependencyGraphProof (EQUIVALENT) 42.99/11.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 122 less nodes. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (6) 42.99/11.90 Obligation: 42.99/11.90 Q DP problem: 42.99/11.90 The TRS P consists of the following rules: 42.99/11.90 42.99/11.90 2^1(4(1(1(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 4^1(x1) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(1(3(4(4(x1)))))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 4^1(1(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(4(4(x1)))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(2(x1)) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 4^1(1(2(x1))) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(4(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(x1) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 1^1(2(x1)) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 4^1(1(2(3(1(x1))))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 1^1(x1) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 4^1(1(2(1(3(x1))))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(2(1(3(x1)))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 2^1(1(3(x1))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(3(x1)) 42.99/11.90 42.99/11.90 The TRS R consists of the following rules: 42.99/11.90 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (7) QDPOrderProof (EQUIVALENT) 42.99/11.90 We use the reduction pair processor [LPAR04,JAR06]. 42.99/11.90 42.99/11.90 42.99/11.90 The following pairs can be oriented strictly and are deleted. 42.99/11.90 42.99/11.90 2^1(1(1(3(5(x1))))) -> 4^1(1(2(1(3(x1))))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(2(1(3(x1)))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 2^1(1(3(x1))) 42.99/11.90 2^1(1(1(3(5(x1))))) -> 1^1(3(x1)) 42.99/11.90 The remaining pairs can at least be oriented weakly. 42.99/11.90 Used ordering: Polynomial interpretation [POLO]: 42.99/11.90 42.99/11.90 POL(0(x_1)) = x_1 42.99/11.90 POL(1(x_1)) = x_1 42.99/11.90 POL(1^1(x_1)) = x_1 42.99/11.90 POL(2(x_1)) = x_1 42.99/11.90 POL(2^1(x_1)) = x_1 42.99/11.90 POL(3(x_1)) = x_1 42.99/11.90 POL(4(x_1)) = x_1 42.99/11.90 POL(4^1(x_1)) = x_1 42.99/11.90 POL(5(x_1)) = 1 + x_1 42.99/11.90 42.99/11.90 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.99/11.90 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 42.99/11.90 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (8) 42.99/11.90 Obligation: 42.99/11.90 Q DP problem: 42.99/11.90 The TRS P consists of the following rules: 42.99/11.90 42.99/11.90 2^1(4(1(1(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 4^1(x1) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(1(3(4(4(x1)))))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 4^1(1(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(4(4(x1)))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(2(x1)) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 4^1(1(2(x1))) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(4(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(x1) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 1^1(2(x1)) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 4^1(1(2(3(1(x1))))) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 1^1(x1) 42.99/11.90 42.99/11.90 The TRS R consists of the following rules: 42.99/11.90 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (9) QDPOrderProof (EQUIVALENT) 42.99/11.90 We use the reduction pair processor [LPAR04,JAR06]. 42.99/11.90 42.99/11.90 42.99/11.90 The following pairs can be oriented strictly and are deleted. 42.99/11.90 42.99/11.90 2^1(4(1(1(0(x1))))) -> 1^1(x1) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(1(3(4(4(x1)))))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(x1) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 1^1(3(4(4(x1)))) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 1^1(2(x1)) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 4^1(1(2(x1))) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(4(x1)) 42.99/11.90 4^1(1(1(3(3(x1))))) -> 4^1(x1) 42.99/11.90 4^1(1(1(0(4(x1))))) -> 1^1(x1) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 1^1(2(x1)) 42.99/11.90 2^1(1(0(4(1(x1))))) -> 2^1(x1) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 1^1(x1) 42.99/11.90 The remaining pairs can at least be oriented weakly. 42.99/11.90 Used ordering: Polynomial interpretation [POLO]: 42.99/11.90 42.99/11.90 POL(0(x_1)) = x_1 42.99/11.90 POL(1(x_1)) = 1 + x_1 42.99/11.90 POL(1^1(x_1)) = x_1 42.99/11.90 POL(2(x_1)) = x_1 42.99/11.90 POL(2^1(x_1)) = x_1 42.99/11.90 POL(3(x_1)) = x_1 42.99/11.90 POL(4(x_1)) = x_1 42.99/11.90 POL(4^1(x_1)) = x_1 42.99/11.90 POL(5(x_1)) = 0 42.99/11.90 42.99/11.90 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.99/11.90 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 42.99/11.90 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (10) 42.99/11.90 Obligation: 42.99/11.90 Q DP problem: 42.99/11.90 The TRS P consists of the following rules: 42.99/11.90 42.99/11.90 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 4^1(x1) 42.99/11.90 1^1(3(2(1(0(x1))))) -> 4^1(1(x1)) 42.99/11.90 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 2^1(1(1(3(3(x1))))) -> 4^1(1(2(3(1(x1))))) 42.99/11.90 42.99/11.90 The TRS R consists of the following rules: 42.99/11.90 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (11) DependencyGraphProof (EQUIVALENT) 42.99/11.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (12) 42.99/11.90 Obligation: 42.99/11.90 Q DP problem: 42.99/11.90 The TRS P consists of the following rules: 42.99/11.90 42.99/11.90 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 42.99/11.90 The TRS R consists of the following rules: 42.99/11.90 42.99/11.90 2(1(1(0(x1)))) -> 2(1(3(0(1(0(x1)))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(0(2(3(0(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 2(4(3(0(1(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 1(2(3(0(4(x1))))) 42.99/11.90 1(3(2(0(x1)))) -> 4(1(2(3(0(0(x1)))))) 42.99/11.90 1(1(0(3(x1)))) -> 0(1(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 4(2(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(3(x1)))) -> 2(1(4(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 4(1(2(3(0(x1))))) 42.99/11.90 1(2(0(3(x1)))) -> 0(1(2(4(3(0(x1)))))) 42.99/11.90 2(1(0(1(0(x1))))) -> 3(1(1(2(0(0(x1)))))) 42.99/11.90 2(4(1(1(0(x1))))) -> 2(4(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(2(1(3(0(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 2(1(3(0(4(1(x1)))))) 42.99/11.90 1(3(2(1(0(x1))))) -> 1(2(3(0(1(2(x1)))))) 42.99/11.90 1(3(5(1(0(x1))))) -> 1(4(5(1(3(0(x1)))))) 42.99/11.90 1(2(3(2(0(x1))))) -> 2(1(4(2(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 0(2(1(4(3(0(x1)))))) 42.99/11.90 1(3(4(2(0(x1))))) -> 2(4(3(0(0(1(x1)))))) 42.99/11.90 2(1(3(3(0(x1))))) -> 2(3(0(1(0(3(x1)))))) 42.99/11.90 1(2(3(3(0(x1))))) -> 4(1(3(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 4(5(1(2(3(0(x1)))))) 42.99/11.90 1(3(2(5(0(x1))))) -> 1(2(3(0(5(0(x1)))))) 42.99/11.90 1(3(4(5(0(x1))))) -> 5(4(1(4(3(0(x1)))))) 42.99/11.90 2(1(0(4(1(x1))))) -> 1(3(0(4(1(2(x1)))))) 42.99/11.90 2(0(1(0(3(x1))))) -> 1(4(0(2(3(0(x1)))))) 42.99/11.90 1(3(2(0(3(x1))))) -> 2(1(3(0(0(3(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(0(5(2(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 4(1(2(5(3(0(x1)))))) 42.99/11.90 1(5(2(0(3(x1))))) -> 1(2(3(0(4(5(x1)))))) 42.99/11.90 1(5(4(0(3(x1))))) -> 1(4(3(0(0(5(x1)))))) 42.99/11.90 4(1(0(3(3(x1))))) -> 4(1(3(0(0(3(x1)))))) 42.99/11.90 1(1(1(3(3(x1))))) -> 1(1(3(1(3(0(x1)))))) 42.99/11.90 2(1(1(3(3(x1))))) -> 3(4(1(2(3(1(x1)))))) 42.99/11.90 4(1(1(3(3(x1))))) -> 1(3(1(3(4(4(x1)))))) 42.99/11.90 1(2(3(4(3(x1))))) -> 1(4(2(3(0(3(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(5(0(2(3(0(x1)))))) 42.99/11.90 1(2(0(5(3(x1))))) -> 1(4(2(5(3(0(x1)))))) 42.99/11.90 4(1(1(0(4(x1))))) -> 1(4(4(3(0(1(x1)))))) 42.99/11.90 1(2(3(5(4(x1))))) -> 4(2(5(1(3(0(x1)))))) 42.99/11.90 4(1(1(0(5(x1))))) -> 4(1(5(1(3(0(x1)))))) 42.99/11.90 1(3(2(0(5(x1))))) -> 0(1(2(3(0(5(x1)))))) 42.99/11.90 1(1(0(3(5(x1))))) -> 0(5(1(1(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 5(1(2(4(3(0(x1)))))) 42.99/11.90 1(2(0(3(5(x1))))) -> 4(2(1(5(3(0(x1)))))) 42.99/11.90 2(1(1(3(5(x1))))) -> 5(4(1(2(1(3(x1)))))) 42.99/11.90 4(1(1(3(5(x1))))) -> 1(2(5(1(4(3(x1)))))) 42.99/11.90 1(4(3(4(5(x1))))) -> 5(1(4(4(3(0(x1)))))) 42.99/11.90 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (13) UsableRulesProof (EQUIVALENT) 42.99/11.90 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (14) 42.99/11.90 Obligation: 42.99/11.90 Q DP problem: 42.99/11.90 The TRS P consists of the following rules: 42.99/11.90 42.99/11.90 1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 42.99/11.90 R is empty. 42.99/11.90 Q is empty. 42.99/11.90 We have to consider all minimal (P,Q,R)-chains. 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (15) QDPSizeChangeProof (EQUIVALENT) 42.99/11.90 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 42.99/11.90 42.99/11.90 From the DPs we obtained the following set of size-change graphs: 42.99/11.90 *1^1(3(4(2(0(x1))))) -> 1^1(x1) 42.99/11.90 The graph contains the following edges 1 > 1 42.99/11.90 42.99/11.90 42.99/11.90 *1^1(3(2(0(x1)))) -> 1^1(x1) 42.99/11.90 The graph contains the following edges 1 > 1 42.99/11.90 42.99/11.90 42.99/11.90 ---------------------------------------- 42.99/11.90 42.99/11.90 (16) 42.99/11.90 YES 43.14/12.01 EOF