14.34/4.91 YES 16.31/5.44 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 16.31/5.44 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 16.31/5.44 16.31/5.44 16.31/5.44 Termination w.r.t. Q of the given QTRS could be proven: 16.31/5.44 16.31/5.44 (0) QTRS 16.31/5.44 (1) DependencyPairsProof [EQUIVALENT, 309 ms] 16.31/5.44 (2) QDP 16.31/5.44 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 16.31/5.44 (4) AND 16.31/5.44 (5) QDP 16.31/5.44 (6) UsableRulesProof [EQUIVALENT, 0 ms] 16.31/5.44 (7) QDP 16.31/5.44 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 16.31/5.44 (9) YES 16.31/5.44 (10) QDP 16.31/5.44 (11) UsableRulesProof [EQUIVALENT, 0 ms] 16.31/5.44 (12) QDP 16.31/5.44 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 16.31/5.44 (14) YES 16.31/5.44 16.31/5.44 16.31/5.44 ---------------------------------------- 16.31/5.44 16.31/5.44 (0) 16.31/5.44 Obligation: 16.31/5.44 Q restricted rewrite system: 16.31/5.44 The TRS R consists of the following rules: 16.31/5.44 16.31/5.44 0(1(2(x1))) -> 0(2(1(1(x1)))) 16.31/5.44 2(3(1(x1))) -> 3(4(2(1(x1)))) 16.31/5.44 0(1(2(0(x1)))) -> 0(0(2(1(1(x1))))) 16.31/5.44 0(1(2(0(x1)))) -> 0(2(1(1(0(x1))))) 16.31/5.44 0(1(2(1(x1)))) -> 0(2(1(1(1(x1))))) 16.31/5.44 0(1(2(4(x1)))) -> 4(0(2(1(1(x1))))) 16.31/5.44 0(1(2(5(x1)))) -> 0(2(5(1(1(x1))))) 16.31/5.44 0(1(2(5(x1)))) -> 0(4(2(1(5(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 0(0(3(1(1(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 3(0(2(1(1(1(x1)))))) 16.31/5.46 0(1(4(1(x1)))) -> 4(0(1(1(1(x1))))) 16.31/5.46 0(1(4(5(x1)))) -> 0(4(1(1(5(x1))))) 16.31/5.46 0(2(0(1(x1)))) -> 0(0(2(4(1(x1))))) 16.31/5.46 0(5(3(1(x1)))) -> 5(0(3(1(1(1(x1)))))) 16.31/5.46 0(5(3(2(x1)))) -> 0(3(4(2(5(x1))))) 16.31/5.46 2(3(2(0(x1)))) -> 2(2(1(3(0(x1))))) 16.31/5.46 2(4(5(2(x1)))) -> 2(1(4(2(5(x1))))) 16.31/5.46 3(0(1(2(x1)))) -> 3(4(0(2(1(x1))))) 16.31/5.46 4(4(3(2(x1)))) -> 4(3(4(2(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 3(5(4(4(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 4(3(1(5(1(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(4(2(5(4(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(5(4(2(1(x1))))) 16.31/5.46 0(1(0(2(2(x1))))) -> 0(0(2(1(4(2(x1)))))) 16.31/5.46 0(1(0(3(1(x1))))) -> 0(3(4(0(1(1(x1)))))) 16.31/5.46 0(1(4(5(1(x1))))) -> 4(0(2(5(1(1(x1)))))) 16.31/5.46 0(1(5(0(1(x1))))) -> 0(0(1(5(5(1(x1)))))) 16.31/5.46 0(2(3(1(3(x1))))) -> 0(3(4(2(1(3(x1)))))) 16.31/5.46 0(4(1(5(2(x1))))) -> 0(4(2(5(1(1(x1)))))) 16.31/5.46 0(5(1(3(2(x1))))) -> 3(0(2(1(1(5(x1)))))) 16.31/5.46 0(5(2(0(4(x1))))) -> 0(4(2(5(0(4(x1)))))) 16.31/5.46 0(5(3(1(5(x1))))) -> 5(0(3(4(1(5(x1)))))) 16.31/5.46 0(5(3(2(5(x1))))) -> 0(4(3(5(2(5(x1)))))) 16.31/5.46 2(0(1(3(1(x1))))) -> 1(1(4(3(0(2(x1)))))) 16.31/5.46 2(0(1(3(5(x1))))) -> 2(5(1(1(0(3(x1)))))) 16.31/5.46 2(0(1(5(3(x1))))) -> 2(1(1(3(0(5(x1)))))) 16.31/5.46 2(3(1(0(1(x1))))) -> 0(3(2(1(1(5(x1)))))) 16.31/5.46 2(3(1(4(1(x1))))) -> 3(4(2(5(1(1(x1)))))) 16.31/5.46 2(3(5(1(2(x1))))) -> 2(3(2(1(5(1(x1)))))) 16.31/5.46 3(0(1(2(0(x1))))) -> 3(4(0(2(1(0(x1)))))) 16.31/5.46 3(2(0(1(0(x1))))) -> 3(2(1(5(0(0(x1)))))) 16.31/5.46 3(2(3(5(1(x1))))) -> 3(3(4(2(1(5(x1)))))) 16.31/5.46 3(5(3(1(3(x1))))) -> 3(5(4(3(1(3(x1)))))) 16.31/5.46 4(0(3(3(1(x1))))) -> 0(3(4(3(1(1(x1)))))) 16.31/5.46 4(4(3(2(5(x1))))) -> 3(4(2(4(1(5(x1)))))) 16.31/5.46 4(5(0(5(2(x1))))) -> 0(4(2(1(5(5(x1)))))) 16.31/5.46 4(5(2(3(1(x1))))) -> 5(3(4(3(2(1(x1)))))) 16.31/5.46 4(5(2(5(2(x1))))) -> 4(2(5(5(2(1(x1)))))) 16.31/5.46 4(5(3(5(1(x1))))) -> 4(5(5(4(3(1(x1)))))) 16.31/5.46 16.31/5.46 Q is empty. 16.31/5.46 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (1) DependencyPairsProof (EQUIVALENT) 16.31/5.46 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (2) 16.31/5.46 Obligation: 16.31/5.46 Q DP problem: 16.31/5.46 The TRS P consists of the following rules: 16.31/5.46 16.31/5.46 0^1(1(2(x1))) -> 0^1(2(1(1(x1)))) 16.31/5.46 0^1(1(2(x1))) -> 2^1(1(1(x1))) 16.31/5.46 2^1(3(1(x1))) -> 3^1(4(2(1(x1)))) 16.31/5.46 2^1(3(1(x1))) -> 4^1(2(1(x1))) 16.31/5.46 2^1(3(1(x1))) -> 2^1(1(x1)) 16.31/5.46 0^1(1(2(0(x1)))) -> 0^1(0(2(1(1(x1))))) 16.31/5.46 0^1(1(2(0(x1)))) -> 0^1(2(1(1(x1)))) 16.31/5.46 0^1(1(2(0(x1)))) -> 2^1(1(1(x1))) 16.31/5.46 0^1(1(2(0(x1)))) -> 0^1(2(1(1(0(x1))))) 16.31/5.46 0^1(1(2(0(x1)))) -> 2^1(1(1(0(x1)))) 16.31/5.46 0^1(1(2(1(x1)))) -> 0^1(2(1(1(1(x1))))) 16.31/5.46 0^1(1(2(1(x1)))) -> 2^1(1(1(1(x1)))) 16.31/5.46 0^1(1(2(4(x1)))) -> 4^1(0(2(1(1(x1))))) 16.31/5.46 0^1(1(2(4(x1)))) -> 0^1(2(1(1(x1)))) 16.31/5.46 0^1(1(2(4(x1)))) -> 2^1(1(1(x1))) 16.31/5.46 0^1(1(2(5(x1)))) -> 0^1(2(5(1(1(x1))))) 16.31/5.46 0^1(1(2(5(x1)))) -> 2^1(5(1(1(x1)))) 16.31/5.46 0^1(1(2(5(x1)))) -> 0^1(4(2(1(5(x1))))) 16.31/5.46 0^1(1(2(5(x1)))) -> 4^1(2(1(5(x1)))) 16.31/5.46 0^1(1(2(5(x1)))) -> 2^1(1(5(x1))) 16.31/5.46 0^1(1(3(1(x1)))) -> 0^1(0(3(1(1(x1))))) 16.31/5.46 0^1(1(3(1(x1)))) -> 0^1(3(1(1(x1)))) 16.31/5.46 0^1(1(3(1(x1)))) -> 3^1(1(1(x1))) 16.31/5.46 0^1(1(3(1(x1)))) -> 3^1(0(2(1(1(1(x1)))))) 16.31/5.46 0^1(1(3(1(x1)))) -> 0^1(2(1(1(1(x1))))) 16.31/5.46 0^1(1(3(1(x1)))) -> 2^1(1(1(1(x1)))) 16.31/5.46 0^1(1(4(1(x1)))) -> 4^1(0(1(1(1(x1))))) 16.31/5.46 0^1(1(4(1(x1)))) -> 0^1(1(1(1(x1)))) 16.31/5.46 0^1(1(4(5(x1)))) -> 0^1(4(1(1(5(x1))))) 16.31/5.46 0^1(1(4(5(x1)))) -> 4^1(1(1(5(x1)))) 16.31/5.46 0^1(2(0(1(x1)))) -> 0^1(0(2(4(1(x1))))) 16.31/5.46 0^1(2(0(1(x1)))) -> 0^1(2(4(1(x1)))) 16.31/5.46 0^1(2(0(1(x1)))) -> 2^1(4(1(x1))) 16.31/5.46 0^1(2(0(1(x1)))) -> 4^1(1(x1)) 16.31/5.46 0^1(5(3(1(x1)))) -> 0^1(3(1(1(1(x1))))) 16.31/5.46 0^1(5(3(1(x1)))) -> 3^1(1(1(1(x1)))) 16.31/5.46 0^1(5(3(2(x1)))) -> 0^1(3(4(2(5(x1))))) 16.31/5.46 0^1(5(3(2(x1)))) -> 3^1(4(2(5(x1)))) 16.31/5.46 0^1(5(3(2(x1)))) -> 4^1(2(5(x1))) 16.31/5.46 0^1(5(3(2(x1)))) -> 2^1(5(x1)) 16.31/5.46 2^1(3(2(0(x1)))) -> 2^1(2(1(3(0(x1))))) 16.31/5.46 2^1(3(2(0(x1)))) -> 2^1(1(3(0(x1)))) 16.31/5.46 2^1(3(2(0(x1)))) -> 3^1(0(x1)) 16.31/5.46 2^1(4(5(2(x1)))) -> 2^1(1(4(2(5(x1))))) 16.31/5.46 2^1(4(5(2(x1)))) -> 4^1(2(5(x1))) 16.31/5.46 2^1(4(5(2(x1)))) -> 2^1(5(x1)) 16.31/5.46 3^1(0(1(2(x1)))) -> 3^1(4(0(2(1(x1))))) 16.31/5.46 3^1(0(1(2(x1)))) -> 4^1(0(2(1(x1)))) 16.31/5.46 3^1(0(1(2(x1)))) -> 0^1(2(1(x1))) 16.31/5.46 3^1(0(1(2(x1)))) -> 2^1(1(x1)) 16.31/5.46 4^1(4(3(2(x1)))) -> 4^1(3(4(2(1(x1))))) 16.31/5.46 4^1(4(3(2(x1)))) -> 3^1(4(2(1(x1)))) 16.31/5.46 4^1(4(3(2(x1)))) -> 4^1(2(1(x1))) 16.31/5.46 4^1(4(3(2(x1)))) -> 2^1(1(x1)) 16.31/5.46 4^1(5(3(1(x1)))) -> 3^1(5(4(4(1(x1))))) 16.31/5.46 4^1(5(3(1(x1)))) -> 4^1(4(1(x1))) 16.31/5.46 4^1(5(3(1(x1)))) -> 4^1(1(x1)) 16.31/5.46 4^1(5(3(1(x1)))) -> 4^1(3(1(5(1(x1))))) 16.31/5.46 4^1(5(3(1(x1)))) -> 3^1(1(5(1(x1)))) 16.31/5.46 4^1(5(3(2(x1)))) -> 3^1(4(2(5(4(x1))))) 16.31/5.46 4^1(5(3(2(x1)))) -> 4^1(2(5(4(x1)))) 16.31/5.46 4^1(5(3(2(x1)))) -> 2^1(5(4(x1))) 16.31/5.46 4^1(5(3(2(x1)))) -> 4^1(x1) 16.31/5.46 4^1(5(3(2(x1)))) -> 3^1(5(4(2(1(x1))))) 16.31/5.46 4^1(5(3(2(x1)))) -> 4^1(2(1(x1))) 16.31/5.46 4^1(5(3(2(x1)))) -> 2^1(1(x1)) 16.31/5.46 0^1(1(0(2(2(x1))))) -> 0^1(0(2(1(4(2(x1)))))) 16.31/5.46 0^1(1(0(2(2(x1))))) -> 0^1(2(1(4(2(x1))))) 16.31/5.46 0^1(1(0(2(2(x1))))) -> 2^1(1(4(2(x1)))) 16.31/5.46 0^1(1(0(2(2(x1))))) -> 4^1(2(x1)) 16.31/5.46 0^1(1(0(3(1(x1))))) -> 0^1(3(4(0(1(1(x1)))))) 16.31/5.46 0^1(1(0(3(1(x1))))) -> 3^1(4(0(1(1(x1))))) 16.31/5.46 0^1(1(0(3(1(x1))))) -> 4^1(0(1(1(x1)))) 16.31/5.46 0^1(1(0(3(1(x1))))) -> 0^1(1(1(x1))) 16.31/5.46 0^1(1(4(5(1(x1))))) -> 4^1(0(2(5(1(1(x1)))))) 16.31/5.46 0^1(1(4(5(1(x1))))) -> 0^1(2(5(1(1(x1))))) 16.31/5.46 0^1(1(4(5(1(x1))))) -> 2^1(5(1(1(x1)))) 16.31/5.46 0^1(1(5(0(1(x1))))) -> 0^1(0(1(5(5(1(x1)))))) 16.31/5.46 0^1(1(5(0(1(x1))))) -> 0^1(1(5(5(1(x1))))) 16.31/5.46 0^1(2(3(1(3(x1))))) -> 0^1(3(4(2(1(3(x1)))))) 16.31/5.46 0^1(2(3(1(3(x1))))) -> 3^1(4(2(1(3(x1))))) 16.31/5.46 0^1(2(3(1(3(x1))))) -> 4^1(2(1(3(x1)))) 16.31/5.46 0^1(2(3(1(3(x1))))) -> 2^1(1(3(x1))) 16.31/5.46 0^1(4(1(5(2(x1))))) -> 0^1(4(2(5(1(1(x1)))))) 16.31/5.46 0^1(4(1(5(2(x1))))) -> 4^1(2(5(1(1(x1))))) 16.31/5.46 0^1(4(1(5(2(x1))))) -> 2^1(5(1(1(x1)))) 16.31/5.46 0^1(5(1(3(2(x1))))) -> 3^1(0(2(1(1(5(x1)))))) 16.31/5.46 0^1(5(1(3(2(x1))))) -> 0^1(2(1(1(5(x1))))) 16.31/5.46 0^1(5(1(3(2(x1))))) -> 2^1(1(1(5(x1)))) 16.31/5.46 0^1(5(2(0(4(x1))))) -> 0^1(4(2(5(0(4(x1)))))) 16.31/5.46 0^1(5(2(0(4(x1))))) -> 4^1(2(5(0(4(x1))))) 16.31/5.46 0^1(5(2(0(4(x1))))) -> 2^1(5(0(4(x1)))) 16.31/5.46 0^1(5(3(1(5(x1))))) -> 0^1(3(4(1(5(x1))))) 16.31/5.46 0^1(5(3(1(5(x1))))) -> 3^1(4(1(5(x1)))) 16.31/5.46 0^1(5(3(1(5(x1))))) -> 4^1(1(5(x1))) 16.31/5.46 0^1(5(3(2(5(x1))))) -> 0^1(4(3(5(2(5(x1)))))) 16.31/5.46 0^1(5(3(2(5(x1))))) -> 4^1(3(5(2(5(x1))))) 16.31/5.46 0^1(5(3(2(5(x1))))) -> 3^1(5(2(5(x1)))) 16.31/5.46 2^1(0(1(3(1(x1))))) -> 4^1(3(0(2(x1)))) 16.31/5.46 2^1(0(1(3(1(x1))))) -> 3^1(0(2(x1))) 16.31/5.46 2^1(0(1(3(1(x1))))) -> 0^1(2(x1)) 16.31/5.46 2^1(0(1(3(1(x1))))) -> 2^1(x1) 16.31/5.46 2^1(0(1(3(5(x1))))) -> 2^1(5(1(1(0(3(x1)))))) 16.31/5.46 2^1(0(1(3(5(x1))))) -> 0^1(3(x1)) 16.31/5.46 2^1(0(1(3(5(x1))))) -> 3^1(x1) 16.31/5.46 2^1(0(1(5(3(x1))))) -> 2^1(1(1(3(0(5(x1)))))) 16.31/5.46 2^1(0(1(5(3(x1))))) -> 3^1(0(5(x1))) 16.31/5.46 2^1(0(1(5(3(x1))))) -> 0^1(5(x1)) 16.31/5.46 2^1(3(1(0(1(x1))))) -> 0^1(3(2(1(1(5(x1)))))) 16.31/5.46 2^1(3(1(0(1(x1))))) -> 3^1(2(1(1(5(x1))))) 16.31/5.46 2^1(3(1(0(1(x1))))) -> 2^1(1(1(5(x1)))) 16.31/5.46 2^1(3(1(4(1(x1))))) -> 3^1(4(2(5(1(1(x1)))))) 16.31/5.46 2^1(3(1(4(1(x1))))) -> 4^1(2(5(1(1(x1))))) 16.31/5.46 2^1(3(1(4(1(x1))))) -> 2^1(5(1(1(x1)))) 16.31/5.46 2^1(3(5(1(2(x1))))) -> 2^1(3(2(1(5(1(x1)))))) 16.31/5.46 2^1(3(5(1(2(x1))))) -> 3^1(2(1(5(1(x1))))) 16.31/5.46 2^1(3(5(1(2(x1))))) -> 2^1(1(5(1(x1)))) 16.31/5.46 3^1(0(1(2(0(x1))))) -> 3^1(4(0(2(1(0(x1)))))) 16.31/5.46 3^1(0(1(2(0(x1))))) -> 4^1(0(2(1(0(x1))))) 16.31/5.46 3^1(0(1(2(0(x1))))) -> 0^1(2(1(0(x1)))) 16.31/5.46 3^1(0(1(2(0(x1))))) -> 2^1(1(0(x1))) 16.31/5.46 3^1(2(0(1(0(x1))))) -> 3^1(2(1(5(0(0(x1)))))) 16.31/5.46 3^1(2(0(1(0(x1))))) -> 2^1(1(5(0(0(x1))))) 16.31/5.46 3^1(2(0(1(0(x1))))) -> 0^1(0(x1)) 16.31/5.46 3^1(2(3(5(1(x1))))) -> 3^1(3(4(2(1(5(x1)))))) 16.31/5.46 3^1(2(3(5(1(x1))))) -> 3^1(4(2(1(5(x1))))) 16.31/5.46 3^1(2(3(5(1(x1))))) -> 4^1(2(1(5(x1)))) 16.31/5.46 3^1(2(3(5(1(x1))))) -> 2^1(1(5(x1))) 16.31/5.46 3^1(5(3(1(3(x1))))) -> 3^1(5(4(3(1(3(x1)))))) 16.31/5.46 3^1(5(3(1(3(x1))))) -> 4^1(3(1(3(x1)))) 16.31/5.46 4^1(0(3(3(1(x1))))) -> 0^1(3(4(3(1(1(x1)))))) 16.31/5.46 4^1(0(3(3(1(x1))))) -> 3^1(4(3(1(1(x1))))) 16.31/5.46 4^1(0(3(3(1(x1))))) -> 4^1(3(1(1(x1)))) 16.31/5.46 4^1(0(3(3(1(x1))))) -> 3^1(1(1(x1))) 16.31/5.46 4^1(4(3(2(5(x1))))) -> 3^1(4(2(4(1(5(x1)))))) 16.31/5.46 4^1(4(3(2(5(x1))))) -> 4^1(2(4(1(5(x1))))) 16.31/5.46 4^1(4(3(2(5(x1))))) -> 2^1(4(1(5(x1)))) 16.31/5.46 4^1(4(3(2(5(x1))))) -> 4^1(1(5(x1))) 16.31/5.46 4^1(5(0(5(2(x1))))) -> 0^1(4(2(1(5(5(x1)))))) 16.31/5.46 4^1(5(0(5(2(x1))))) -> 4^1(2(1(5(5(x1))))) 16.31/5.46 4^1(5(0(5(2(x1))))) -> 2^1(1(5(5(x1)))) 16.31/5.46 4^1(5(2(3(1(x1))))) -> 3^1(4(3(2(1(x1))))) 16.31/5.46 4^1(5(2(3(1(x1))))) -> 4^1(3(2(1(x1)))) 16.31/5.46 4^1(5(2(3(1(x1))))) -> 3^1(2(1(x1))) 16.31/5.46 4^1(5(2(3(1(x1))))) -> 2^1(1(x1)) 16.31/5.46 4^1(5(2(5(2(x1))))) -> 4^1(2(5(5(2(1(x1)))))) 16.31/5.46 4^1(5(2(5(2(x1))))) -> 2^1(5(5(2(1(x1))))) 16.31/5.46 4^1(5(2(5(2(x1))))) -> 2^1(1(x1)) 16.31/5.46 4^1(5(3(5(1(x1))))) -> 4^1(5(5(4(3(1(x1)))))) 16.31/5.46 4^1(5(3(5(1(x1))))) -> 4^1(3(1(x1))) 16.31/5.46 4^1(5(3(5(1(x1))))) -> 3^1(1(x1)) 16.31/5.46 16.31/5.46 The TRS R consists of the following rules: 16.31/5.46 16.31/5.46 0(1(2(x1))) -> 0(2(1(1(x1)))) 16.31/5.46 2(3(1(x1))) -> 3(4(2(1(x1)))) 16.31/5.46 0(1(2(0(x1)))) -> 0(0(2(1(1(x1))))) 16.31/5.46 0(1(2(0(x1)))) -> 0(2(1(1(0(x1))))) 16.31/5.46 0(1(2(1(x1)))) -> 0(2(1(1(1(x1))))) 16.31/5.46 0(1(2(4(x1)))) -> 4(0(2(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(2(5(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(4(2(1(5(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 0(0(3(1(1(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 3(0(2(1(1(1(x1)))))) 16.31/5.46 0(1(4(1(x1)))) -> 4(0(1(1(1(x1))))) 16.31/5.46 0(1(4(5(x1)))) -> 0(4(1(1(5(x1))))) 16.31/5.46 0(2(0(1(x1)))) -> 0(0(2(4(1(x1))))) 16.31/5.46 0(5(3(1(x1)))) -> 5(0(3(1(1(1(x1)))))) 16.31/5.46 0(5(3(2(x1)))) -> 0(3(4(2(5(x1))))) 16.31/5.46 2(3(2(0(x1)))) -> 2(2(1(3(0(x1))))) 16.31/5.46 2(4(5(2(x1)))) -> 2(1(4(2(5(x1))))) 16.31/5.46 3(0(1(2(x1)))) -> 3(4(0(2(1(x1))))) 16.31/5.46 4(4(3(2(x1)))) -> 4(3(4(2(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 3(5(4(4(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 4(3(1(5(1(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(4(2(5(4(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(5(4(2(1(x1))))) 16.31/5.46 0(1(0(2(2(x1))))) -> 0(0(2(1(4(2(x1)))))) 16.31/5.46 0(1(0(3(1(x1))))) -> 0(3(4(0(1(1(x1)))))) 16.31/5.46 0(1(4(5(1(x1))))) -> 4(0(2(5(1(1(x1)))))) 16.31/5.46 0(1(5(0(1(x1))))) -> 0(0(1(5(5(1(x1)))))) 16.31/5.46 0(2(3(1(3(x1))))) -> 0(3(4(2(1(3(x1)))))) 16.31/5.46 0(4(1(5(2(x1))))) -> 0(4(2(5(1(1(x1)))))) 16.31/5.46 0(5(1(3(2(x1))))) -> 3(0(2(1(1(5(x1)))))) 16.31/5.46 0(5(2(0(4(x1))))) -> 0(4(2(5(0(4(x1)))))) 16.31/5.46 0(5(3(1(5(x1))))) -> 5(0(3(4(1(5(x1)))))) 16.31/5.46 0(5(3(2(5(x1))))) -> 0(4(3(5(2(5(x1)))))) 16.31/5.46 2(0(1(3(1(x1))))) -> 1(1(4(3(0(2(x1)))))) 16.31/5.46 2(0(1(3(5(x1))))) -> 2(5(1(1(0(3(x1)))))) 16.31/5.46 2(0(1(5(3(x1))))) -> 2(1(1(3(0(5(x1)))))) 16.31/5.46 2(3(1(0(1(x1))))) -> 0(3(2(1(1(5(x1)))))) 16.31/5.46 2(3(1(4(1(x1))))) -> 3(4(2(5(1(1(x1)))))) 16.31/5.46 2(3(5(1(2(x1))))) -> 2(3(2(1(5(1(x1)))))) 16.31/5.46 3(0(1(2(0(x1))))) -> 3(4(0(2(1(0(x1)))))) 16.31/5.46 3(2(0(1(0(x1))))) -> 3(2(1(5(0(0(x1)))))) 16.31/5.46 3(2(3(5(1(x1))))) -> 3(3(4(2(1(5(x1)))))) 16.31/5.46 3(5(3(1(3(x1))))) -> 3(5(4(3(1(3(x1)))))) 16.31/5.46 4(0(3(3(1(x1))))) -> 0(3(4(3(1(1(x1)))))) 16.31/5.46 4(4(3(2(5(x1))))) -> 3(4(2(4(1(5(x1)))))) 16.31/5.46 4(5(0(5(2(x1))))) -> 0(4(2(1(5(5(x1)))))) 16.31/5.46 4(5(2(3(1(x1))))) -> 5(3(4(3(2(1(x1)))))) 16.31/5.46 4(5(2(5(2(x1))))) -> 4(2(5(5(2(1(x1)))))) 16.31/5.46 4(5(3(5(1(x1))))) -> 4(5(5(4(3(1(x1)))))) 16.31/5.46 16.31/5.46 Q is empty. 16.31/5.46 We have to consider all minimal (P,Q,R)-chains. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (3) DependencyGraphProof (EQUIVALENT) 16.31/5.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 149 less nodes. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (4) 16.31/5.46 Complex Obligation (AND) 16.31/5.46 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (5) 16.31/5.46 Obligation: 16.31/5.46 Q DP problem: 16.31/5.46 The TRS P consists of the following rules: 16.31/5.46 16.31/5.46 4^1(5(3(2(x1)))) -> 4^1(x1) 16.31/5.46 16.31/5.46 The TRS R consists of the following rules: 16.31/5.46 16.31/5.46 0(1(2(x1))) -> 0(2(1(1(x1)))) 16.31/5.46 2(3(1(x1))) -> 3(4(2(1(x1)))) 16.31/5.46 0(1(2(0(x1)))) -> 0(0(2(1(1(x1))))) 16.31/5.46 0(1(2(0(x1)))) -> 0(2(1(1(0(x1))))) 16.31/5.46 0(1(2(1(x1)))) -> 0(2(1(1(1(x1))))) 16.31/5.46 0(1(2(4(x1)))) -> 4(0(2(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(2(5(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(4(2(1(5(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 0(0(3(1(1(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 3(0(2(1(1(1(x1)))))) 16.31/5.46 0(1(4(1(x1)))) -> 4(0(1(1(1(x1))))) 16.31/5.46 0(1(4(5(x1)))) -> 0(4(1(1(5(x1))))) 16.31/5.46 0(2(0(1(x1)))) -> 0(0(2(4(1(x1))))) 16.31/5.46 0(5(3(1(x1)))) -> 5(0(3(1(1(1(x1)))))) 16.31/5.46 0(5(3(2(x1)))) -> 0(3(4(2(5(x1))))) 16.31/5.46 2(3(2(0(x1)))) -> 2(2(1(3(0(x1))))) 16.31/5.46 2(4(5(2(x1)))) -> 2(1(4(2(5(x1))))) 16.31/5.46 3(0(1(2(x1)))) -> 3(4(0(2(1(x1))))) 16.31/5.46 4(4(3(2(x1)))) -> 4(3(4(2(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 3(5(4(4(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 4(3(1(5(1(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(4(2(5(4(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(5(4(2(1(x1))))) 16.31/5.46 0(1(0(2(2(x1))))) -> 0(0(2(1(4(2(x1)))))) 16.31/5.46 0(1(0(3(1(x1))))) -> 0(3(4(0(1(1(x1)))))) 16.31/5.46 0(1(4(5(1(x1))))) -> 4(0(2(5(1(1(x1)))))) 16.31/5.46 0(1(5(0(1(x1))))) -> 0(0(1(5(5(1(x1)))))) 16.31/5.46 0(2(3(1(3(x1))))) -> 0(3(4(2(1(3(x1)))))) 16.31/5.46 0(4(1(5(2(x1))))) -> 0(4(2(5(1(1(x1)))))) 16.31/5.46 0(5(1(3(2(x1))))) -> 3(0(2(1(1(5(x1)))))) 16.31/5.46 0(5(2(0(4(x1))))) -> 0(4(2(5(0(4(x1)))))) 16.31/5.46 0(5(3(1(5(x1))))) -> 5(0(3(4(1(5(x1)))))) 16.31/5.46 0(5(3(2(5(x1))))) -> 0(4(3(5(2(5(x1)))))) 16.31/5.46 2(0(1(3(1(x1))))) -> 1(1(4(3(0(2(x1)))))) 16.31/5.46 2(0(1(3(5(x1))))) -> 2(5(1(1(0(3(x1)))))) 16.31/5.46 2(0(1(5(3(x1))))) -> 2(1(1(3(0(5(x1)))))) 16.31/5.46 2(3(1(0(1(x1))))) -> 0(3(2(1(1(5(x1)))))) 16.31/5.46 2(3(1(4(1(x1))))) -> 3(4(2(5(1(1(x1)))))) 16.31/5.46 2(3(5(1(2(x1))))) -> 2(3(2(1(5(1(x1)))))) 16.31/5.46 3(0(1(2(0(x1))))) -> 3(4(0(2(1(0(x1)))))) 16.31/5.46 3(2(0(1(0(x1))))) -> 3(2(1(5(0(0(x1)))))) 16.31/5.46 3(2(3(5(1(x1))))) -> 3(3(4(2(1(5(x1)))))) 16.31/5.46 3(5(3(1(3(x1))))) -> 3(5(4(3(1(3(x1)))))) 16.31/5.46 4(0(3(3(1(x1))))) -> 0(3(4(3(1(1(x1)))))) 16.31/5.46 4(4(3(2(5(x1))))) -> 3(4(2(4(1(5(x1)))))) 16.31/5.46 4(5(0(5(2(x1))))) -> 0(4(2(1(5(5(x1)))))) 16.31/5.46 4(5(2(3(1(x1))))) -> 5(3(4(3(2(1(x1)))))) 16.31/5.46 4(5(2(5(2(x1))))) -> 4(2(5(5(2(1(x1)))))) 16.31/5.46 4(5(3(5(1(x1))))) -> 4(5(5(4(3(1(x1)))))) 16.31/5.46 16.31/5.46 Q is empty. 16.31/5.46 We have to consider all minimal (P,Q,R)-chains. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (6) UsableRulesProof (EQUIVALENT) 16.31/5.46 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. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (7) 16.31/5.46 Obligation: 16.31/5.46 Q DP problem: 16.31/5.46 The TRS P consists of the following rules: 16.31/5.46 16.31/5.46 4^1(5(3(2(x1)))) -> 4^1(x1) 16.31/5.46 16.31/5.46 R is empty. 16.31/5.46 Q is empty. 16.31/5.46 We have to consider all minimal (P,Q,R)-chains. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (8) QDPSizeChangeProof (EQUIVALENT) 16.31/5.46 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. 16.31/5.46 16.31/5.46 From the DPs we obtained the following set of size-change graphs: 16.31/5.46 *4^1(5(3(2(x1)))) -> 4^1(x1) 16.31/5.46 The graph contains the following edges 1 > 1 16.31/5.46 16.31/5.46 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (9) 16.31/5.46 YES 16.31/5.46 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (10) 16.31/5.46 Obligation: 16.31/5.46 Q DP problem: 16.31/5.46 The TRS P consists of the following rules: 16.31/5.46 16.31/5.46 2^1(0(1(3(1(x1))))) -> 2^1(x1) 16.31/5.46 16.31/5.46 The TRS R consists of the following rules: 16.31/5.46 16.31/5.46 0(1(2(x1))) -> 0(2(1(1(x1)))) 16.31/5.46 2(3(1(x1))) -> 3(4(2(1(x1)))) 16.31/5.46 0(1(2(0(x1)))) -> 0(0(2(1(1(x1))))) 16.31/5.46 0(1(2(0(x1)))) -> 0(2(1(1(0(x1))))) 16.31/5.46 0(1(2(1(x1)))) -> 0(2(1(1(1(x1))))) 16.31/5.46 0(1(2(4(x1)))) -> 4(0(2(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(2(5(1(1(x1))))) 16.31/5.46 0(1(2(5(x1)))) -> 0(4(2(1(5(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 0(0(3(1(1(x1))))) 16.31/5.46 0(1(3(1(x1)))) -> 3(0(2(1(1(1(x1)))))) 16.31/5.46 0(1(4(1(x1)))) -> 4(0(1(1(1(x1))))) 16.31/5.46 0(1(4(5(x1)))) -> 0(4(1(1(5(x1))))) 16.31/5.46 0(2(0(1(x1)))) -> 0(0(2(4(1(x1))))) 16.31/5.46 0(5(3(1(x1)))) -> 5(0(3(1(1(1(x1)))))) 16.31/5.46 0(5(3(2(x1)))) -> 0(3(4(2(5(x1))))) 16.31/5.46 2(3(2(0(x1)))) -> 2(2(1(3(0(x1))))) 16.31/5.46 2(4(5(2(x1)))) -> 2(1(4(2(5(x1))))) 16.31/5.46 3(0(1(2(x1)))) -> 3(4(0(2(1(x1))))) 16.31/5.46 4(4(3(2(x1)))) -> 4(3(4(2(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 3(5(4(4(1(x1))))) 16.31/5.46 4(5(3(1(x1)))) -> 4(3(1(5(1(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(4(2(5(4(x1))))) 16.31/5.46 4(5(3(2(x1)))) -> 3(5(4(2(1(x1))))) 16.31/5.46 0(1(0(2(2(x1))))) -> 0(0(2(1(4(2(x1)))))) 16.31/5.46 0(1(0(3(1(x1))))) -> 0(3(4(0(1(1(x1)))))) 16.31/5.46 0(1(4(5(1(x1))))) -> 4(0(2(5(1(1(x1)))))) 16.31/5.46 0(1(5(0(1(x1))))) -> 0(0(1(5(5(1(x1)))))) 16.31/5.46 0(2(3(1(3(x1))))) -> 0(3(4(2(1(3(x1)))))) 16.31/5.46 0(4(1(5(2(x1))))) -> 0(4(2(5(1(1(x1)))))) 16.31/5.46 0(5(1(3(2(x1))))) -> 3(0(2(1(1(5(x1)))))) 16.31/5.46 0(5(2(0(4(x1))))) -> 0(4(2(5(0(4(x1)))))) 16.31/5.46 0(5(3(1(5(x1))))) -> 5(0(3(4(1(5(x1)))))) 16.31/5.46 0(5(3(2(5(x1))))) -> 0(4(3(5(2(5(x1)))))) 16.31/5.46 2(0(1(3(1(x1))))) -> 1(1(4(3(0(2(x1)))))) 16.31/5.46 2(0(1(3(5(x1))))) -> 2(5(1(1(0(3(x1)))))) 16.31/5.46 2(0(1(5(3(x1))))) -> 2(1(1(3(0(5(x1)))))) 16.31/5.46 2(3(1(0(1(x1))))) -> 0(3(2(1(1(5(x1)))))) 16.31/5.46 2(3(1(4(1(x1))))) -> 3(4(2(5(1(1(x1)))))) 16.31/5.46 2(3(5(1(2(x1))))) -> 2(3(2(1(5(1(x1)))))) 16.31/5.46 3(0(1(2(0(x1))))) -> 3(4(0(2(1(0(x1)))))) 16.31/5.46 3(2(0(1(0(x1))))) -> 3(2(1(5(0(0(x1)))))) 16.31/5.46 3(2(3(5(1(x1))))) -> 3(3(4(2(1(5(x1)))))) 16.31/5.46 3(5(3(1(3(x1))))) -> 3(5(4(3(1(3(x1)))))) 16.31/5.46 4(0(3(3(1(x1))))) -> 0(3(4(3(1(1(x1)))))) 16.31/5.46 4(4(3(2(5(x1))))) -> 3(4(2(4(1(5(x1)))))) 16.31/5.46 4(5(0(5(2(x1))))) -> 0(4(2(1(5(5(x1)))))) 16.31/5.46 4(5(2(3(1(x1))))) -> 5(3(4(3(2(1(x1)))))) 16.31/5.46 4(5(2(5(2(x1))))) -> 4(2(5(5(2(1(x1)))))) 16.31/5.46 4(5(3(5(1(x1))))) -> 4(5(5(4(3(1(x1)))))) 16.31/5.46 16.31/5.46 Q is empty. 16.31/5.46 We have to consider all minimal (P,Q,R)-chains. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (11) UsableRulesProof (EQUIVALENT) 16.31/5.46 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. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (12) 16.31/5.46 Obligation: 16.31/5.46 Q DP problem: 16.31/5.46 The TRS P consists of the following rules: 16.31/5.46 16.31/5.46 2^1(0(1(3(1(x1))))) -> 2^1(x1) 16.31/5.46 16.31/5.46 R is empty. 16.31/5.46 Q is empty. 16.31/5.46 We have to consider all minimal (P,Q,R)-chains. 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (13) QDPSizeChangeProof (EQUIVALENT) 16.31/5.46 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. 16.31/5.46 16.31/5.46 From the DPs we obtained the following set of size-change graphs: 16.31/5.46 *2^1(0(1(3(1(x1))))) -> 2^1(x1) 16.31/5.46 The graph contains the following edges 1 > 1 16.31/5.46 16.31/5.46 16.31/5.46 ---------------------------------------- 16.31/5.46 16.31/5.46 (14) 16.31/5.46 YES 16.60/9.42 EOF