27.31/7.75 YES 27.66/7.83 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 27.66/7.83 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 27.66/7.83 27.66/7.83 27.66/7.83 Termination w.r.t. Q of the given QTRS could be proven: 27.66/7.83 27.66/7.83 (0) QTRS 27.66/7.83 (1) DependencyPairsProof [EQUIVALENT, 184 ms] 27.66/7.83 (2) QDP 27.66/7.83 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 27.66/7.83 (4) QDP 27.66/7.83 (5) QDPOrderProof [EQUIVALENT, 255 ms] 27.66/7.83 (6) QDP 27.66/7.83 (7) DependencyGraphProof [EQUIVALENT, 0 ms] 27.66/7.83 (8) TRUE 27.66/7.83 27.66/7.83 27.66/7.83 ---------------------------------------- 27.66/7.83 27.66/7.83 (0) 27.66/7.83 Obligation: 27.66/7.83 Q restricted rewrite system: 27.66/7.83 The TRS R consists of the following rules: 27.66/7.83 27.66/7.83 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 27.66/7.83 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 27.66/7.83 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 27.66/7.83 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 27.66/7.83 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 27.66/7.83 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 27.66/7.83 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 27.66/7.83 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 27.66/7.83 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 27.66/7.83 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 27.66/7.83 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 27.66/7.83 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 27.66/7.83 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 27.66/7.83 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 27.66/7.83 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 27.66/7.83 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 27.66/7.83 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 27.66/7.83 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 27.66/7.83 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 27.66/7.83 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 27.66/7.83 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 27.66/7.83 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 27.66/7.83 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 27.66/7.83 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 27.66/7.83 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 27.66/7.83 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 27.66/7.83 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 27.66/7.83 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 27.66/7.83 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 27.66/7.83 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) 27.66/7.83 27.66/7.83 Q is empty. 27.66/7.83 27.66/7.83 ---------------------------------------- 27.66/7.83 27.66/7.83 (1) DependencyPairsProof (EQUIVALENT) 27.66/7.83 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 27.66/7.83 ---------------------------------------- 27.66/7.83 27.66/7.83 (2) 27.66/7.83 Obligation: 27.66/7.83 Q DP problem: 27.66/7.83 The TRS P consists of the following rules: 27.66/7.83 27.66/7.83 2^1(5(x1)) -> 1^1(3(3(0(1(0(x1)))))) 27.66/7.83 2^1(5(x1)) -> 3^1(3(0(1(0(x1))))) 27.66/7.83 2^1(5(x1)) -> 3^1(0(1(0(x1)))) 27.66/7.83 2^1(5(x1)) -> 1^1(0(x1)) 27.66/7.83 2^1(5(x1)) -> 2^1(2(0(5(0(1(x1)))))) 27.66/7.83 2^1(5(x1)) -> 2^1(0(5(0(1(x1))))) 27.66/7.83 2^1(5(x1)) -> 5^1(0(1(x1))) 27.66/7.83 2^1(5(x1)) -> 1^1(x1) 27.66/7.83 3^1(5(x1)) -> 1^1(3(2(0(0(1(x1)))))) 27.66/7.83 3^1(5(x1)) -> 3^1(2(0(0(1(x1))))) 27.66/7.83 3^1(5(x1)) -> 2^1(0(0(1(x1)))) 27.66/7.83 3^1(5(x1)) -> 1^1(x1) 27.66/7.83 3^1(5(x1)) -> 3^1(2(0(5(3(0(x1)))))) 27.66/7.83 3^1(5(x1)) -> 2^1(0(5(3(0(x1))))) 27.66/7.83 3^1(5(x1)) -> 5^1(3(0(x1))) 27.66/7.83 3^1(5(x1)) -> 3^1(0(x1)) 27.66/7.83 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 27.66/7.83 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 27.66/7.83 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 27.66/7.83 4^1(5(x1)) -> 3^1(2(1(x1))) 27.66/7.83 4^1(5(x1)) -> 2^1(1(x1)) 27.66/7.83 4^1(5(x1)) -> 1^1(x1) 27.66/7.83 4^1(5(x1)) -> 3^1(2(0(5(0(0(x1)))))) 27.66/7.83 4^1(5(x1)) -> 2^1(0(5(0(0(x1))))) 27.66/7.83 4^1(5(x1)) -> 5^1(0(0(x1))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(0(5(0(5(4(x1)))))) 27.66/7.83 1^1(2(5(x1))) -> 5^1(0(5(4(x1)))) 27.66/7.83 1^1(2(5(x1))) -> 5^1(4(x1)) 27.66/7.83 1^1(2(5(x1))) -> 4^1(x1) 27.66/7.83 1^1(2(5(x1))) -> 1^1(2(2(1(0(1(x1)))))) 27.66/7.83 1^1(2(5(x1))) -> 2^1(2(1(0(1(x1))))) 27.66/7.83 1^1(2(5(x1))) -> 2^1(1(0(1(x1)))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(0(1(x1))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(x1) 27.66/7.83 1^1(2(5(x1))) -> 2^1(0(1(3(1(0(x1)))))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(3(1(0(x1)))) 27.66/7.83 1^1(2(5(x1))) -> 3^1(1(0(x1))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(0(x1)) 27.66/7.83 1^1(4(5(x1))) -> 1^1(2(4(0(2(1(x1)))))) 27.66/7.83 1^1(4(5(x1))) -> 2^1(4(0(2(1(x1))))) 27.66/7.83 1^1(4(5(x1))) -> 4^1(0(2(1(x1)))) 27.66/7.83 1^1(4(5(x1))) -> 2^1(1(x1)) 27.66/7.83 1^1(4(5(x1))) -> 1^1(x1) 27.66/7.83 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 27.66/7.83 2^1(5(1(x1))) -> 1^1(2(3(x1))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(3(x1)) 27.66/7.83 2^1(5(1(x1))) -> 3^1(x1) 27.66/7.83 2^1(5(2(x1))) -> 4^1(0(2(2(3(3(x1)))))) 27.66/7.83 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 27.66/7.83 2^1(5(2(x1))) -> 2^1(3(3(x1))) 27.66/7.83 2^1(5(2(x1))) -> 3^1(3(x1)) 27.66/7.83 2^1(5(2(x1))) -> 3^1(x1) 27.66/7.83 2^1(5(3(x1))) -> 2^1(0(4(1(3(3(x1)))))) 27.66/7.83 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 27.66/7.83 2^1(5(3(x1))) -> 1^1(3(3(x1))) 27.66/7.83 2^1(5(3(x1))) -> 3^1(3(x1)) 27.66/7.83 2^1(5(4(x1))) -> 2^1(0(5(1(0(1(x1)))))) 27.66/7.83 2^1(5(4(x1))) -> 5^1(1(0(1(x1)))) 27.66/7.83 2^1(5(4(x1))) -> 1^1(0(1(x1))) 27.66/7.83 2^1(5(4(x1))) -> 1^1(x1) 27.66/7.83 3^1(2(5(x1))) -> 3^1(2(0(1(0(5(x1)))))) 27.66/7.83 3^1(2(5(x1))) -> 2^1(0(1(0(5(x1))))) 27.66/7.83 3^1(2(5(x1))) -> 1^1(0(5(x1))) 27.66/7.83 3^1(4(2(x1))) -> 3^1(4(0(2(2(2(x1)))))) 27.66/7.83 3^1(4(2(x1))) -> 4^1(0(2(2(2(x1))))) 27.66/7.83 3^1(4(2(x1))) -> 2^1(2(2(x1))) 27.66/7.83 3^1(4(2(x1))) -> 2^1(2(x1)) 27.66/7.83 3^1(5(1(x1))) -> 4^1(2(0(0(5(x1))))) 27.66/7.83 3^1(5(1(x1))) -> 2^1(0(0(5(x1)))) 27.66/7.83 3^1(5(1(x1))) -> 5^1(x1) 27.66/7.83 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 27.66/7.83 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 27.66/7.83 3^1(5(1(x1))) -> 2^1(3(4(x1))) 27.66/7.83 3^1(5(1(x1))) -> 3^1(4(x1)) 27.66/7.83 3^1(5(1(x1))) -> 4^1(x1) 27.66/7.83 3^1(5(1(x1))) -> 2^1(1(4(1(0(1(x1)))))) 27.66/7.83 3^1(5(1(x1))) -> 1^1(4(1(0(1(x1))))) 27.66/7.83 3^1(5(1(x1))) -> 4^1(1(0(1(x1)))) 27.66/7.83 3^1(5(1(x1))) -> 1^1(0(1(x1))) 27.66/7.83 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(2(2(x1))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(2(x1)) 27.66/7.83 3^1(5(2(x1))) -> 2^1(0(2(2(3(0(x1)))))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(2(3(0(x1)))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(3(0(x1))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(0(x1)) 27.66/7.83 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(1(2(x1))) 27.66/7.83 3^1(5(2(x1))) -> 1^1(2(x1)) 27.66/7.83 3^1(5(3(x1))) -> 2^1(4(3(3(0(x1))))) 27.66/7.83 3^1(5(3(x1))) -> 4^1(3(3(0(x1)))) 27.66/7.83 3^1(5(3(x1))) -> 3^1(3(0(x1))) 27.66/7.83 3^1(5(3(x1))) -> 3^1(0(x1)) 27.66/7.83 3^1(5(3(x1))) -> 5^1(4(3(3(0(x1))))) 27.66/7.83 3^1(5(3(x1))) -> 2^1(3(4(0(4(2(x1)))))) 27.66/7.83 3^1(5(3(x1))) -> 3^1(4(0(4(2(x1))))) 27.66/7.83 3^1(5(3(x1))) -> 4^1(0(4(2(x1)))) 27.66/7.83 3^1(5(3(x1))) -> 4^1(2(x1)) 27.66/7.83 3^1(5(3(x1))) -> 2^1(x1) 27.66/7.83 3^1(5(4(x1))) -> 2^1(0(5(0(0(x1))))) 27.66/7.83 3^1(5(4(x1))) -> 5^1(0(0(x1))) 27.66/7.83 3^1(5(4(x1))) -> 5^1(0(0(1(2(x1))))) 27.66/7.83 3^1(5(4(x1))) -> 1^1(2(x1)) 27.66/7.83 3^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 3^1(5(5(x1))) -> 5^1(4(1(0(5(x1))))) 27.66/7.83 3^1(5(5(x1))) -> 4^1(1(0(5(x1)))) 27.66/7.83 3^1(5(5(x1))) -> 1^1(0(5(x1))) 27.66/7.83 4^1(5(1(x1))) -> 2^1(1(0(5(3(3(x1)))))) 27.66/7.83 4^1(5(1(x1))) -> 1^1(0(5(3(3(x1))))) 27.66/7.83 4^1(5(1(x1))) -> 5^1(3(3(x1))) 27.66/7.83 4^1(5(1(x1))) -> 3^1(3(x1)) 27.66/7.83 4^1(5(1(x1))) -> 3^1(x1) 27.66/7.83 4^1(5(2(x1))) -> 5^1(1(0(0(4(x1))))) 27.66/7.83 4^1(5(2(x1))) -> 1^1(0(0(4(x1)))) 27.66/7.83 4^1(5(2(x1))) -> 4^1(x1) 27.66/7.83 4^1(5(4(x1))) -> 2^1(2(1(0(4(2(x1)))))) 27.66/7.83 4^1(5(4(x1))) -> 2^1(1(0(4(2(x1))))) 27.66/7.83 4^1(5(4(x1))) -> 1^1(0(4(2(x1)))) 27.66/7.83 4^1(5(4(x1))) -> 4^1(2(x1)) 27.66/7.83 4^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 4^1(5(4(x1))) -> 3^1(2(0(3(2(0(x1)))))) 27.66/7.83 4^1(5(4(x1))) -> 2^1(0(3(2(0(x1))))) 27.66/7.83 4^1(5(4(x1))) -> 3^1(2(0(x1))) 27.66/7.83 4^1(5(4(x1))) -> 2^1(0(x1)) 27.66/7.83 5^1(5(3(x1))) -> 5^1(1(0(1(2(2(x1)))))) 27.66/7.83 5^1(5(3(x1))) -> 1^1(0(1(2(2(x1))))) 27.66/7.83 5^1(5(3(x1))) -> 1^1(2(2(x1))) 27.66/7.83 5^1(5(3(x1))) -> 2^1(2(x1)) 27.66/7.83 5^1(5(3(x1))) -> 2^1(x1) 27.66/7.83 5^1(5(4(x1))) -> 5^1(1(0(4(2(2(x1)))))) 27.66/7.83 5^1(5(4(x1))) -> 1^1(0(4(2(2(x1))))) 27.66/7.83 5^1(5(4(x1))) -> 4^1(2(2(x1))) 27.66/7.83 5^1(5(4(x1))) -> 2^1(2(x1)) 27.66/7.83 5^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 27.66/7.83 The TRS R consists of the following rules: 27.66/7.83 27.66/7.83 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 27.66/7.83 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 27.66/7.83 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 27.66/7.83 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 27.66/7.83 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 27.66/7.83 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 27.66/7.83 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 27.66/7.83 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 27.66/7.83 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 27.66/7.83 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 27.66/7.83 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 27.66/7.83 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 27.66/7.83 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 27.66/7.83 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 27.66/7.83 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 27.66/7.83 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 27.66/7.83 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 27.66/7.83 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 27.66/7.83 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 27.66/7.83 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 27.66/7.83 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 27.66/7.83 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 27.66/7.83 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 27.66/7.83 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 27.66/7.83 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 27.66/7.83 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 27.66/7.83 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 27.66/7.83 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 27.66/7.83 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 27.66/7.83 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) 27.66/7.83 27.66/7.83 Q is empty. 27.66/7.83 We have to consider all minimal (P,Q,R)-chains. 27.66/7.83 ---------------------------------------- 27.66/7.83 27.66/7.83 (3) DependencyGraphProof (EQUIVALENT) 27.66/7.83 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 79 less nodes. 27.66/7.83 ---------------------------------------- 27.66/7.83 27.66/7.83 (4) 27.66/7.83 Obligation: 27.66/7.83 Q DP problem: 27.66/7.83 The TRS P consists of the following rules: 27.66/7.83 27.66/7.83 2^1(5(x1)) -> 1^1(x1) 27.66/7.83 1^1(2(5(x1))) -> 5^1(4(x1)) 27.66/7.83 5^1(5(3(x1))) -> 1^1(2(2(x1))) 27.66/7.83 1^1(2(5(x1))) -> 4^1(x1) 27.66/7.83 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 27.66/7.83 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 27.66/7.83 2^1(5(1(x1))) -> 1^1(2(3(x1))) 27.66/7.83 1^1(2(5(x1))) -> 1^1(x1) 27.66/7.83 1^1(4(5(x1))) -> 2^1(1(x1)) 27.66/7.83 2^1(5(1(x1))) -> 2^1(3(x1)) 27.66/7.83 2^1(5(1(x1))) -> 3^1(x1) 27.66/7.83 3^1(5(x1)) -> 1^1(x1) 27.66/7.83 1^1(4(5(x1))) -> 1^1(x1) 27.66/7.83 3^1(4(2(x1))) -> 2^1(2(2(x1))) 27.66/7.83 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 27.66/7.83 2^1(5(2(x1))) -> 2^1(3(3(x1))) 27.66/7.83 2^1(5(2(x1))) -> 3^1(3(x1)) 27.66/7.83 3^1(4(2(x1))) -> 2^1(2(x1)) 27.66/7.83 2^1(5(2(x1))) -> 3^1(x1) 27.66/7.83 3^1(5(1(x1))) -> 5^1(x1) 27.66/7.83 5^1(5(3(x1))) -> 2^1(2(x1)) 27.66/7.83 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 27.66/7.83 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 27.66/7.83 2^1(5(3(x1))) -> 1^1(3(3(x1))) 27.66/7.83 2^1(5(3(x1))) -> 3^1(3(x1)) 27.66/7.83 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 27.66/7.83 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 27.66/7.83 4^1(5(x1)) -> 3^1(2(1(x1))) 27.66/7.83 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 27.66/7.83 2^1(5(4(x1))) -> 1^1(x1) 27.66/7.83 3^1(5(1(x1))) -> 2^1(3(4(x1))) 27.66/7.83 3^1(5(1(x1))) -> 3^1(4(x1)) 27.66/7.83 3^1(5(1(x1))) -> 4^1(x1) 27.66/7.83 4^1(5(x1)) -> 2^1(1(x1)) 27.66/7.83 4^1(5(x1)) -> 1^1(x1) 27.66/7.83 4^1(5(1(x1))) -> 5^1(3(3(x1))) 27.66/7.83 5^1(5(3(x1))) -> 2^1(x1) 27.66/7.83 5^1(5(4(x1))) -> 4^1(2(2(x1))) 27.66/7.83 4^1(5(1(x1))) -> 3^1(3(x1)) 27.66/7.83 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 27.66/7.83 4^1(5(1(x1))) -> 3^1(x1) 27.66/7.83 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(2(2(x1))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(2(x1)) 27.66/7.83 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 27.66/7.83 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 27.66/7.83 3^1(5(2(x1))) -> 2^1(1(2(x1))) 27.66/7.83 3^1(5(2(x1))) -> 1^1(2(x1)) 27.66/7.83 3^1(5(3(x1))) -> 4^1(2(x1)) 27.66/7.83 4^1(5(2(x1))) -> 4^1(x1) 27.66/7.83 4^1(5(4(x1))) -> 4^1(2(x1)) 27.66/7.83 4^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 3^1(5(3(x1))) -> 2^1(x1) 27.66/7.83 3^1(5(4(x1))) -> 1^1(2(x1)) 27.66/7.83 3^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 5^1(5(4(x1))) -> 2^1(2(x1)) 27.66/7.83 5^1(5(4(x1))) -> 2^1(x1) 27.66/7.83 27.66/7.83 The TRS R consists of the following rules: 27.66/7.83 27.66/7.83 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 27.66/7.83 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 27.66/7.83 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 27.66/7.83 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 27.66/7.83 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 27.66/7.83 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 27.66/7.83 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 27.66/7.83 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 27.66/7.83 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 27.66/7.83 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 27.66/7.83 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 27.66/7.84 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 27.66/7.84 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 27.66/7.84 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 27.66/7.84 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 27.66/7.84 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 27.66/7.84 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 27.66/7.84 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 27.66/7.84 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 27.66/7.84 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 27.66/7.84 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 27.66/7.84 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 27.66/7.84 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) 27.66/7.84 27.66/7.84 Q is empty. 27.66/7.84 We have to consider all minimal (P,Q,R)-chains. 27.66/7.84 ---------------------------------------- 27.66/7.84 27.66/7.84 (5) QDPOrderProof (EQUIVALENT) 27.66/7.84 We use the reduction pair processor [LPAR04,JAR06]. 27.66/7.84 27.66/7.84 27.66/7.84 The following pairs can be oriented strictly and are deleted. 27.66/7.84 27.66/7.84 2^1(5(x1)) -> 1^1(x1) 27.66/7.84 1^1(2(5(x1))) -> 5^1(4(x1)) 27.66/7.84 5^1(5(3(x1))) -> 1^1(2(2(x1))) 27.66/7.84 1^1(2(5(x1))) -> 4^1(x1) 27.66/7.84 4^1(5(x1)) -> 2^1(2(1(3(2(1(x1)))))) 27.66/7.84 2^1(5(1(x1))) -> 2^1(2(2(1(2(3(x1)))))) 27.66/7.84 2^1(5(1(x1))) -> 2^1(2(1(2(3(x1))))) 27.66/7.84 2^1(5(1(x1))) -> 2^1(1(2(3(x1)))) 27.66/7.84 2^1(5(1(x1))) -> 1^1(2(3(x1))) 27.66/7.84 1^1(2(5(x1))) -> 1^1(x1) 27.66/7.84 1^1(4(5(x1))) -> 2^1(1(x1)) 27.66/7.84 2^1(5(1(x1))) -> 2^1(3(x1)) 27.66/7.84 2^1(5(1(x1))) -> 3^1(x1) 27.66/7.84 3^1(5(x1)) -> 1^1(x1) 27.66/7.84 1^1(4(5(x1))) -> 1^1(x1) 27.66/7.84 2^1(5(2(x1))) -> 2^1(2(3(3(x1)))) 27.66/7.84 2^1(5(2(x1))) -> 2^1(3(3(x1))) 27.66/7.84 2^1(5(2(x1))) -> 3^1(3(x1)) 27.66/7.84 2^1(5(2(x1))) -> 3^1(x1) 27.66/7.84 3^1(5(1(x1))) -> 5^1(x1) 27.66/7.84 5^1(5(3(x1))) -> 2^1(2(x1)) 27.66/7.84 2^1(5(3(x1))) -> 4^1(1(3(3(x1)))) 27.66/7.84 4^1(5(x1)) -> 2^1(1(3(2(1(x1))))) 27.66/7.84 2^1(5(3(x1))) -> 1^1(3(3(x1))) 27.66/7.84 2^1(5(3(x1))) -> 3^1(3(x1)) 27.66/7.84 3^1(5(1(x1))) -> 4^1(2(2(3(4(x1))))) 27.66/7.84 4^1(5(x1)) -> 1^1(3(2(1(x1)))) 27.66/7.84 4^1(5(x1)) -> 3^1(2(1(x1))) 27.66/7.84 3^1(5(1(x1))) -> 2^1(2(3(4(x1)))) 27.66/7.84 2^1(5(4(x1))) -> 1^1(x1) 27.66/7.84 3^1(5(1(x1))) -> 2^1(3(4(x1))) 27.66/7.84 3^1(5(1(x1))) -> 3^1(4(x1)) 27.66/7.84 3^1(5(1(x1))) -> 4^1(x1) 27.66/7.84 4^1(5(x1)) -> 2^1(1(x1)) 27.66/7.84 4^1(5(x1)) -> 1^1(x1) 27.66/7.84 4^1(5(1(x1))) -> 5^1(3(3(x1))) 27.66/7.84 5^1(5(3(x1))) -> 2^1(x1) 27.66/7.84 5^1(5(4(x1))) -> 4^1(2(2(x1))) 27.66/7.84 4^1(5(1(x1))) -> 3^1(3(x1)) 27.66/7.84 3^1(5(2(x1))) -> 4^1(3(2(2(2(x1))))) 27.66/7.84 4^1(5(1(x1))) -> 3^1(x1) 27.66/7.84 3^1(5(2(x1))) -> 3^1(2(2(2(x1)))) 27.66/7.84 3^1(5(2(x1))) -> 2^1(2(2(x1))) 27.66/7.84 3^1(5(2(x1))) -> 2^1(2(x1)) 27.66/7.84 3^1(5(2(x1))) -> 2^1(3(3(2(1(2(x1)))))) 27.66/7.84 3^1(5(2(x1))) -> 3^1(3(2(1(2(x1))))) 27.66/7.84 3^1(5(2(x1))) -> 3^1(2(1(2(x1)))) 27.66/7.84 3^1(5(2(x1))) -> 2^1(1(2(x1))) 27.66/7.84 3^1(5(2(x1))) -> 1^1(2(x1)) 27.66/7.84 3^1(5(3(x1))) -> 4^1(2(x1)) 27.66/7.84 4^1(5(2(x1))) -> 4^1(x1) 27.66/7.84 4^1(5(4(x1))) -> 4^1(2(x1)) 27.66/7.84 4^1(5(4(x1))) -> 2^1(x1) 27.66/7.84 3^1(5(3(x1))) -> 2^1(x1) 27.66/7.84 3^1(5(4(x1))) -> 1^1(2(x1)) 27.66/7.84 3^1(5(4(x1))) -> 2^1(x1) 27.66/7.84 5^1(5(4(x1))) -> 2^1(2(x1)) 27.66/7.84 5^1(5(4(x1))) -> 2^1(x1) 27.66/7.84 The remaining pairs can at least be oriented weakly. 27.66/7.84 Used ordering: Polynomial interpretation [POLO]: 27.66/7.84 27.66/7.84 POL(0(x_1)) = 0 27.66/7.84 POL(1(x_1)) = x_1 27.66/7.84 POL(1^1(x_1)) = x_1 27.66/7.84 POL(2(x_1)) = x_1 27.66/7.84 POL(2^1(x_1)) = x_1 27.66/7.84 POL(3(x_1)) = x_1 27.66/7.84 POL(3^1(x_1)) = x_1 27.66/7.84 POL(4(x_1)) = x_1 27.66/7.84 POL(4^1(x_1)) = x_1 27.66/7.84 POL(5(x_1)) = 1 + x_1 27.66/7.84 POL(5^1(x_1)) = x_1 27.66/7.84 27.66/7.84 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 27.66/7.84 27.66/7.84 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 27.66/7.84 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 27.66/7.84 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 27.66/7.84 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 27.66/7.84 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 27.66/7.84 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 27.66/7.84 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 27.66/7.84 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 27.66/7.84 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 27.66/7.84 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 27.66/7.84 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 27.66/7.84 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 27.66/7.84 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 27.66/7.84 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 27.66/7.84 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 27.66/7.84 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 27.66/7.84 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 27.66/7.84 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 27.66/7.84 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 27.66/7.84 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 27.66/7.84 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 27.66/7.84 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 27.66/7.84 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 27.66/7.84 27.66/7.84 27.66/7.84 ---------------------------------------- 27.66/7.84 27.66/7.84 (6) 27.66/7.84 Obligation: 27.66/7.84 Q DP problem: 27.66/7.84 The TRS P consists of the following rules: 27.66/7.84 27.66/7.84 3^1(4(2(x1))) -> 2^1(2(2(x1))) 27.66/7.84 3^1(4(2(x1))) -> 2^1(2(x1)) 27.66/7.84 27.66/7.84 The TRS R consists of the following rules: 27.66/7.84 27.66/7.84 2(5(x1)) -> 1(3(3(0(1(0(x1)))))) 27.66/7.84 2(5(x1)) -> 2(2(0(5(0(1(x1)))))) 27.66/7.84 3(5(x1)) -> 1(3(2(0(0(1(x1)))))) 27.66/7.84 3(5(x1)) -> 3(2(0(5(3(0(x1)))))) 27.66/7.84 4(5(x1)) -> 2(2(1(3(2(1(x1)))))) 27.66/7.84 4(5(x1)) -> 3(2(0(5(0(0(x1)))))) 27.66/7.84 1(2(5(x1))) -> 1(0(5(0(5(4(x1)))))) 27.66/7.84 1(2(5(x1))) -> 1(2(2(1(0(1(x1)))))) 27.66/7.84 1(2(5(x1))) -> 2(0(1(3(1(0(x1)))))) 27.66/7.84 1(4(5(x1))) -> 1(2(4(0(2(1(x1)))))) 27.66/7.84 2(5(1(x1))) -> 2(2(2(1(2(3(x1)))))) 27.66/7.84 2(5(2(x1))) -> 4(0(2(2(3(3(x1)))))) 27.66/7.84 2(5(3(x1))) -> 2(0(4(1(3(3(x1)))))) 27.66/7.84 2(5(4(x1))) -> 2(0(5(1(0(1(x1)))))) 27.66/7.84 3(2(5(x1))) -> 3(2(0(1(0(5(x1)))))) 27.66/7.84 3(4(2(x1))) -> 3(4(0(2(2(2(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(0(0(5(x1)))))) 27.66/7.84 3(5(1(x1))) -> 0(4(2(2(3(4(x1)))))) 27.66/7.84 3(5(1(x1))) -> 2(1(4(1(0(1(x1)))))) 27.66/7.84 3(5(2(x1))) -> 0(4(3(2(2(2(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(0(2(2(3(0(x1)))))) 27.66/7.84 3(5(2(x1))) -> 2(3(3(2(1(2(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(2(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 0(5(4(3(3(0(x1)))))) 27.66/7.84 3(5(3(x1))) -> 2(3(4(0(4(2(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(2(0(5(0(0(x1)))))) 27.66/7.84 3(5(4(x1))) -> 0(5(0(0(1(2(x1)))))) 27.66/7.84 3(5(5(x1))) -> 0(5(4(1(0(5(x1)))))) 27.66/7.84 4(5(1(x1))) -> 2(1(0(5(3(3(x1)))))) 27.66/7.84 4(5(2(x1))) -> 0(5(1(0(0(4(x1)))))) 27.66/7.84 4(5(4(x1))) -> 2(2(1(0(4(2(x1)))))) 27.66/7.84 4(5(4(x1))) -> 3(2(0(3(2(0(x1)))))) 27.66/7.84 5(5(3(x1))) -> 5(1(0(1(2(2(x1)))))) 27.66/7.84 5(5(4(x1))) -> 5(1(0(4(2(2(x1)))))) 27.66/7.84 27.66/7.84 Q is empty. 27.66/7.84 We have to consider all minimal (P,Q,R)-chains. 27.66/7.84 ---------------------------------------- 27.66/7.84 27.66/7.84 (7) DependencyGraphProof (EQUIVALENT) 27.66/7.84 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 27.66/7.84 ---------------------------------------- 27.66/7.84 27.66/7.84 (8) 27.66/7.84 TRUE 27.88/7.97 EOF