33.44/9.37 YES 34.24/9.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 34.24/9.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 34.24/9.58 34.24/9.58 34.24/9.58 Termination w.r.t. Q of the given QTRS could be proven: 34.24/9.58 34.24/9.58 (0) QTRS 34.24/9.58 (1) DependencyPairsProof [EQUIVALENT, 188 ms] 34.24/9.58 (2) QDP 34.24/9.58 (3) DependencyGraphProof [EQUIVALENT, 6 ms] 34.24/9.58 (4) AND 34.24/9.58 (5) QDP 34.24/9.58 (6) UsableRulesProof [EQUIVALENT, 7 ms] 34.24/9.58 (7) QDP 34.24/9.58 (8) QDPOrderProof [EQUIVALENT, 76 ms] 34.24/9.58 (9) QDP 34.24/9.58 (10) DependencyGraphProof [EQUIVALENT, 0 ms] 34.24/9.58 (11) QDP 34.24/9.58 (12) QDPOrderProof [EQUIVALENT, 30 ms] 34.24/9.58 (13) QDP 34.24/9.58 (14) UsableRulesProof [EQUIVALENT, 0 ms] 34.24/9.58 (15) QDP 34.24/9.58 (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] 34.24/9.58 (17) YES 34.24/9.58 (18) QDP 34.24/9.58 (19) UsableRulesProof [EQUIVALENT, 1 ms] 34.24/9.58 (20) QDP 34.24/9.58 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 34.24/9.58 (22) TRUE 34.24/9.58 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (0) 34.24/9.58 Obligation: 34.24/9.58 Q restricted rewrite system: 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 0(1(1(x1))) -> 0(2(1(1(x1)))) 34.24/9.58 0(1(1(x1))) -> 0(0(2(1(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 0(2(1(2(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 2(1(0(2(1(x1))))) 34.24/9.58 3(0(1(x1))) -> 1(3(0(0(2(4(x1)))))) 34.24/9.58 3(5(1(x1))) -> 1(3(4(5(x1)))) 34.24/9.58 3(5(1(x1))) -> 2(4(5(3(1(x1))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 0(1(0(1(x1)))) -> 1(1(0(0(2(4(x1)))))) 34.24/9.58 0(1(2(3(x1)))) -> 3(1(5(0(2(x1))))) 34.24/9.58 0(1(2(3(x1)))) -> 0(3(1(2(1(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 0(2(1(2(4(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 4(0(0(2(1(1(x1)))))) 34.24/9.58 0(1(5(1(x1)))) -> 0(0(2(1(1(5(x1)))))) 34.24/9.58 0(4(5(1(x1)))) -> 0(1(3(4(5(x1))))) 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(3(4(5(1(x1))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(5(3(1(2(x1))))) 34.24/9.58 3(5(1(3(x1)))) -> 3(5(3(1(2(x1))))) 34.24/9.58 3(5(4(1(x1)))) -> 4(1(3(4(5(x1))))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 0(1(2(0(1(x1))))) -> 0(3(0(2(1(1(x1)))))) 34.24/9.58 0(1(2(2(1(x1))))) -> 0(2(1(2(1(3(x1)))))) 34.24/9.58 0(3(0(5(1(x1))))) -> 0(3(4(5(0(1(x1)))))) 34.24/9.58 0(3(4(2(3(x1))))) -> 0(5(3(4(3(2(x1)))))) 34.24/9.58 0(3(5(4(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 0(3(2(4(5(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 4(3(1(0(0(2(x1)))))) 34.24/9.58 0(4(5(5(1(x1))))) -> 2(4(5(5(0(1(x1)))))) 34.24/9.58 0(5(1(0(1(x1))))) -> 0(1(5(5(0(1(x1)))))) 34.24/9.58 0(5(3(2(1(x1))))) -> 0(0(2(5(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 0(2(3(4(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(1(3(3(0(2(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(2(3(3(0(2(x1)))))) 34.24/9.58 3(0(4(1(1(x1))))) -> 0(0(1(3(4(1(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 3(3(4(1(1(x1))))) -> 1(3(4(5(3(1(x1)))))) 34.24/9.58 3(3(5(1(1(x1))))) -> 3(1(4(5(3(1(x1)))))) 34.24/9.58 3(5(4(1(3(x1))))) -> 1(4(5(3(1(3(x1)))))) 34.24/9.58 3(5(4(4(1(x1))))) -> 4(1(4(3(4(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (1) DependencyPairsProof (EQUIVALENT) 34.24/9.58 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (2) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 0^1(1(1(x1))) -> 0^1(2(1(1(x1)))) 34.24/9.58 0^1(1(1(x1))) -> 0^1(0(2(1(1(x1))))) 34.24/9.58 0^1(1(1(x1))) -> 0^1(2(1(2(1(x1))))) 34.24/9.58 0^1(1(1(x1))) -> 0^1(2(1(x1))) 34.24/9.58 3^1(0(1(x1))) -> 3^1(0(0(2(4(x1))))) 34.24/9.58 3^1(0(1(x1))) -> 0^1(0(2(4(x1)))) 34.24/9.58 3^1(0(1(x1))) -> 0^1(2(4(x1))) 34.24/9.58 3^1(5(1(x1))) -> 3^1(4(5(x1))) 34.24/9.58 3^1(5(1(x1))) -> 5^1(x1) 34.24/9.58 3^1(5(1(x1))) -> 5^1(3(1(x1))) 34.24/9.58 3^1(5(1(x1))) -> 3^1(1(x1)) 34.24/9.58 5^1(1(3(x1))) -> 5^1(3(1(2(x1)))) 34.24/9.58 5^1(1(3(x1))) -> 3^1(1(2(x1))) 34.24/9.58 0^1(1(0(1(x1)))) -> 0^1(0(2(4(x1)))) 34.24/9.58 0^1(1(0(1(x1)))) -> 0^1(2(4(x1))) 34.24/9.58 0^1(1(2(3(x1)))) -> 3^1(1(5(0(2(x1))))) 34.24/9.58 0^1(1(2(3(x1)))) -> 5^1(0(2(x1))) 34.24/9.58 0^1(1(2(3(x1)))) -> 0^1(2(x1)) 34.24/9.58 0^1(1(2(3(x1)))) -> 0^1(3(1(2(1(1(x1)))))) 34.24/9.58 0^1(1(2(3(x1)))) -> 3^1(1(2(1(1(x1))))) 34.24/9.58 0^1(1(4(1(x1)))) -> 0^1(2(1(2(4(1(x1)))))) 34.24/9.58 0^1(1(4(1(x1)))) -> 0^1(0(2(1(1(x1))))) 34.24/9.58 0^1(1(4(1(x1)))) -> 0^1(2(1(1(x1)))) 34.24/9.58 0^1(1(5(1(x1)))) -> 0^1(0(2(1(1(5(x1)))))) 34.24/9.58 0^1(1(5(1(x1)))) -> 0^1(2(1(1(5(x1))))) 34.24/9.58 0^1(1(5(1(x1)))) -> 5^1(x1) 34.24/9.58 0^1(4(5(1(x1)))) -> 0^1(1(3(4(5(x1))))) 34.24/9.58 0^1(4(5(1(x1)))) -> 3^1(4(5(x1))) 34.24/9.58 0^1(4(5(1(x1)))) -> 5^1(x1) 34.24/9.58 3^1(2(0(1(x1)))) -> 3^1(0(2(4(5(x1))))) 34.24/9.58 3^1(2(0(1(x1)))) -> 0^1(2(4(5(x1)))) 34.24/9.58 3^1(2(0(1(x1)))) -> 5^1(x1) 34.24/9.58 3^1(5(1(1(x1)))) -> 3^1(4(5(1(x1)))) 34.24/9.58 3^1(5(1(1(x1)))) -> 5^1(1(x1)) 34.24/9.58 3^1(5(1(1(x1)))) -> 5^1(3(1(2(x1)))) 34.24/9.58 3^1(5(1(1(x1)))) -> 3^1(1(2(x1))) 34.24/9.58 3^1(5(1(3(x1)))) -> 3^1(5(3(1(2(x1))))) 34.24/9.58 3^1(5(1(3(x1)))) -> 5^1(3(1(2(x1)))) 34.24/9.58 3^1(5(1(3(x1)))) -> 3^1(1(2(x1))) 34.24/9.58 3^1(5(4(1(x1)))) -> 3^1(4(5(x1))) 34.24/9.58 3^1(5(4(1(x1)))) -> 5^1(x1) 34.24/9.58 5^1(1(2(3(x1)))) -> 5^1(5(3(1(2(x1))))) 34.24/9.58 5^1(1(2(3(x1)))) -> 5^1(3(1(2(x1)))) 34.24/9.58 5^1(1(2(3(x1)))) -> 3^1(1(2(x1))) 34.24/9.58 5^1(2(0(1(x1)))) -> 5^1(3(1(0(2(4(x1)))))) 34.24/9.58 5^1(2(0(1(x1)))) -> 3^1(1(0(2(4(x1))))) 34.24/9.58 5^1(2(0(1(x1)))) -> 0^1(2(4(x1))) 34.24/9.58 5^1(4(3(3(x1)))) -> 3^1(1(3(4(5(x1))))) 34.24/9.58 5^1(4(3(3(x1)))) -> 3^1(4(5(x1))) 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 5^1(5(0(1(x1)))) -> 0^1(2(4(5(5(x1))))) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(5(x1)) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(x1) 34.24/9.58 0^1(1(2(0(1(x1))))) -> 0^1(3(0(2(1(1(x1)))))) 34.24/9.58 0^1(1(2(0(1(x1))))) -> 3^1(0(2(1(1(x1))))) 34.24/9.58 0^1(1(2(0(1(x1))))) -> 0^1(2(1(1(x1)))) 34.24/9.58 0^1(1(2(2(1(x1))))) -> 0^1(2(1(2(1(3(x1)))))) 34.24/9.58 0^1(1(2(2(1(x1))))) -> 3^1(x1) 34.24/9.58 0^1(3(0(5(1(x1))))) -> 0^1(3(4(5(0(1(x1)))))) 34.24/9.58 0^1(3(0(5(1(x1))))) -> 3^1(4(5(0(1(x1))))) 34.24/9.58 0^1(3(0(5(1(x1))))) -> 5^1(0(1(x1))) 34.24/9.58 0^1(3(0(5(1(x1))))) -> 0^1(1(x1)) 34.24/9.58 0^1(3(4(2(3(x1))))) -> 0^1(5(3(4(3(2(x1)))))) 34.24/9.58 0^1(3(4(2(3(x1))))) -> 5^1(3(4(3(2(x1))))) 34.24/9.58 0^1(3(4(2(3(x1))))) -> 3^1(4(3(2(x1)))) 34.24/9.58 0^1(3(4(2(3(x1))))) -> 3^1(2(x1)) 34.24/9.58 0^1(3(5(4(1(x1))))) -> 0^1(5(2(4(3(1(x1)))))) 34.24/9.58 0^1(3(5(4(1(x1))))) -> 5^1(2(4(3(1(x1))))) 34.24/9.58 0^1(3(5(4(1(x1))))) -> 3^1(1(x1)) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 0^1(3(2(4(5(1(x1)))))) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 3^1(2(4(5(1(x1))))) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 5^1(1(x1)) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 3^1(1(0(0(2(x1))))) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 0^1(0(2(x1))) 34.24/9.58 0^1(4(1(2(3(x1))))) -> 0^1(2(x1)) 34.24/9.58 0^1(4(5(5(1(x1))))) -> 5^1(5(0(1(x1)))) 34.24/9.58 0^1(4(5(5(1(x1))))) -> 5^1(0(1(x1))) 34.24/9.58 0^1(4(5(5(1(x1))))) -> 0^1(1(x1)) 34.24/9.58 0^1(5(1(0(1(x1))))) -> 0^1(1(5(5(0(1(x1)))))) 34.24/9.58 0^1(5(1(0(1(x1))))) -> 5^1(5(0(1(x1)))) 34.24/9.58 0^1(5(1(0(1(x1))))) -> 5^1(0(1(x1))) 34.24/9.58 0^1(5(3(2(1(x1))))) -> 0^1(0(2(5(3(1(x1)))))) 34.24/9.58 0^1(5(3(2(1(x1))))) -> 0^1(2(5(3(1(x1))))) 34.24/9.58 0^1(5(3(2(1(x1))))) -> 5^1(3(1(x1))) 34.24/9.58 0^1(5(3(2(1(x1))))) -> 3^1(1(x1)) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 0^1(2(3(4(3(1(x1)))))) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 3^1(4(3(1(x1)))) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 3^1(1(x1)) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 3^1(3(0(2(x1)))) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 3^1(0(2(x1))) 34.24/9.58 3^1(0(1(2(3(x1))))) -> 0^1(2(x1)) 34.24/9.58 3^1(0(4(1(1(x1))))) -> 0^1(0(1(3(4(1(x1)))))) 34.24/9.58 3^1(0(4(1(1(x1))))) -> 0^1(1(3(4(1(x1))))) 34.24/9.58 3^1(0(4(1(1(x1))))) -> 3^1(4(1(x1))) 34.24/9.58 3^1(2(4(1(3(x1))))) -> 3^1(4(3(1(2(x1))))) 34.24/9.58 3^1(2(4(1(3(x1))))) -> 3^1(1(2(x1))) 34.24/9.58 3^1(3(4(1(1(x1))))) -> 3^1(4(5(3(1(x1))))) 34.24/9.58 3^1(3(4(1(1(x1))))) -> 5^1(3(1(x1))) 34.24/9.58 3^1(3(4(1(1(x1))))) -> 3^1(1(x1)) 34.24/9.58 3^1(3(5(1(1(x1))))) -> 3^1(1(4(5(3(1(x1)))))) 34.24/9.58 3^1(3(5(1(1(x1))))) -> 5^1(3(1(x1))) 34.24/9.58 3^1(3(5(1(1(x1))))) -> 3^1(1(x1)) 34.24/9.58 3^1(5(4(1(3(x1))))) -> 5^1(3(1(3(x1)))) 34.24/9.58 3^1(5(4(1(3(x1))))) -> 3^1(1(3(x1))) 34.24/9.58 3^1(5(4(4(1(x1))))) -> 3^1(4(5(x1))) 34.24/9.58 3^1(5(4(4(1(x1))))) -> 5^1(x1) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 3^1(2(4(5(3(2(x1)))))) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 3^1(2(x1)) 34.24/9.58 5^1(4(2(0(1(x1))))) -> 5^1(1(2(0(2(4(x1)))))) 34.24/9.58 5^1(4(2(0(1(x1))))) -> 0^1(2(4(x1))) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 0(1(1(x1))) -> 0(2(1(1(x1)))) 34.24/9.58 0(1(1(x1))) -> 0(0(2(1(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 0(2(1(2(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 2(1(0(2(1(x1))))) 34.24/9.58 3(0(1(x1))) -> 1(3(0(0(2(4(x1)))))) 34.24/9.58 3(5(1(x1))) -> 1(3(4(5(x1)))) 34.24/9.58 3(5(1(x1))) -> 2(4(5(3(1(x1))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 0(1(0(1(x1)))) -> 1(1(0(0(2(4(x1)))))) 34.24/9.58 0(1(2(3(x1)))) -> 3(1(5(0(2(x1))))) 34.24/9.58 0(1(2(3(x1)))) -> 0(3(1(2(1(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 0(2(1(2(4(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 4(0(0(2(1(1(x1)))))) 34.24/9.58 0(1(5(1(x1)))) -> 0(0(2(1(1(5(x1)))))) 34.24/9.58 0(4(5(1(x1)))) -> 0(1(3(4(5(x1))))) 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(3(4(5(1(x1))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(5(3(1(2(x1))))) 34.24/9.58 3(5(1(3(x1)))) -> 3(5(3(1(2(x1))))) 34.24/9.58 3(5(4(1(x1)))) -> 4(1(3(4(5(x1))))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 0(1(2(0(1(x1))))) -> 0(3(0(2(1(1(x1)))))) 34.24/9.58 0(1(2(2(1(x1))))) -> 0(2(1(2(1(3(x1)))))) 34.24/9.58 0(3(0(5(1(x1))))) -> 0(3(4(5(0(1(x1)))))) 34.24/9.58 0(3(4(2(3(x1))))) -> 0(5(3(4(3(2(x1)))))) 34.24/9.58 0(3(5(4(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 0(3(2(4(5(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 4(3(1(0(0(2(x1)))))) 34.24/9.58 0(4(5(5(1(x1))))) -> 2(4(5(5(0(1(x1)))))) 34.24/9.58 0(5(1(0(1(x1))))) -> 0(1(5(5(0(1(x1)))))) 34.24/9.58 0(5(3(2(1(x1))))) -> 0(0(2(5(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 0(2(3(4(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(1(3(3(0(2(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(2(3(3(0(2(x1)))))) 34.24/9.58 3(0(4(1(1(x1))))) -> 0(0(1(3(4(1(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 3(3(4(1(1(x1))))) -> 1(3(4(5(3(1(x1)))))) 34.24/9.58 3(3(5(1(1(x1))))) -> 3(1(4(5(3(1(x1)))))) 34.24/9.58 3(5(4(1(3(x1))))) -> 1(4(5(3(1(3(x1)))))) 34.24/9.58 3(5(4(4(1(x1))))) -> 4(1(4(3(4(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (3) DependencyGraphProof (EQUIVALENT) 34.24/9.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 104 less nodes. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (4) 34.24/9.58 Complex Obligation (AND) 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (5) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(5(x1)) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(x1) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 3^1(2(x1)) 34.24/9.58 3^1(2(0(1(x1)))) -> 5^1(x1) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 0(1(1(x1))) -> 0(2(1(1(x1)))) 34.24/9.58 0(1(1(x1))) -> 0(0(2(1(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 0(2(1(2(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 2(1(0(2(1(x1))))) 34.24/9.58 3(0(1(x1))) -> 1(3(0(0(2(4(x1)))))) 34.24/9.58 3(5(1(x1))) -> 1(3(4(5(x1)))) 34.24/9.58 3(5(1(x1))) -> 2(4(5(3(1(x1))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 0(1(0(1(x1)))) -> 1(1(0(0(2(4(x1)))))) 34.24/9.58 0(1(2(3(x1)))) -> 3(1(5(0(2(x1))))) 34.24/9.58 0(1(2(3(x1)))) -> 0(3(1(2(1(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 0(2(1(2(4(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 4(0(0(2(1(1(x1)))))) 34.24/9.58 0(1(5(1(x1)))) -> 0(0(2(1(1(5(x1)))))) 34.24/9.58 0(4(5(1(x1)))) -> 0(1(3(4(5(x1))))) 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(3(4(5(1(x1))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(5(3(1(2(x1))))) 34.24/9.58 3(5(1(3(x1)))) -> 3(5(3(1(2(x1))))) 34.24/9.58 3(5(4(1(x1)))) -> 4(1(3(4(5(x1))))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 0(1(2(0(1(x1))))) -> 0(3(0(2(1(1(x1)))))) 34.24/9.58 0(1(2(2(1(x1))))) -> 0(2(1(2(1(3(x1)))))) 34.24/9.58 0(3(0(5(1(x1))))) -> 0(3(4(5(0(1(x1)))))) 34.24/9.58 0(3(4(2(3(x1))))) -> 0(5(3(4(3(2(x1)))))) 34.24/9.58 0(3(5(4(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 0(3(2(4(5(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 4(3(1(0(0(2(x1)))))) 34.24/9.58 0(4(5(5(1(x1))))) -> 2(4(5(5(0(1(x1)))))) 34.24/9.58 0(5(1(0(1(x1))))) -> 0(1(5(5(0(1(x1)))))) 34.24/9.58 0(5(3(2(1(x1))))) -> 0(0(2(5(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 0(2(3(4(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(1(3(3(0(2(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(2(3(3(0(2(x1)))))) 34.24/9.58 3(0(4(1(1(x1))))) -> 0(0(1(3(4(1(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 3(3(4(1(1(x1))))) -> 1(3(4(5(3(1(x1)))))) 34.24/9.58 3(3(5(1(1(x1))))) -> 3(1(4(5(3(1(x1)))))) 34.24/9.58 3(5(4(1(3(x1))))) -> 1(4(5(3(1(3(x1)))))) 34.24/9.58 3(5(4(4(1(x1))))) -> 4(1(4(3(4(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (6) UsableRulesProof (EQUIVALENT) 34.24/9.58 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. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (7) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(5(x1)) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(x1) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 3^1(2(x1)) 34.24/9.58 3^1(2(0(1(x1)))) -> 5^1(x1) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (8) QDPOrderProof (EQUIVALENT) 34.24/9.58 We use the reduction pair processor [LPAR04,JAR06]. 34.24/9.58 34.24/9.58 34.24/9.58 The following pairs can be oriented strictly and are deleted. 34.24/9.58 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(5(x1)) 34.24/9.58 5^1(5(0(1(x1)))) -> 5^1(x1) 34.24/9.58 3^1(2(0(1(x1)))) -> 5^1(x1) 34.24/9.58 The remaining pairs can at least be oriented weakly. 34.24/9.58 Used ordering: Polynomial interpretation [POLO]: 34.24/9.58 34.24/9.58 POL(0(x_1)) = 1 + x_1 34.24/9.58 POL(1(x_1)) = x_1 34.24/9.58 POL(2(x_1)) = x_1 34.24/9.58 POL(3(x_1)) = x_1 34.24/9.58 POL(3^1(x_1)) = 1 + x_1 34.24/9.58 POL(4(x_1)) = x_1 34.24/9.58 POL(5(x_1)) = x_1 34.24/9.58 POL(5^1(x_1)) = 1 + x_1 34.24/9.58 34.24/9.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.24/9.58 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (9) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 5^1(2(4(2(3(x1))))) -> 3^1(2(x1)) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (10) DependencyGraphProof (EQUIVALENT) 34.24/9.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (11) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (12) QDPOrderProof (EQUIVALENT) 34.24/9.58 We use the reduction pair processor [LPAR04,JAR06]. 34.24/9.58 34.24/9.58 34.24/9.58 The following pairs can be oriented strictly and are deleted. 34.24/9.58 34.24/9.58 5^1(2(4(2(3(x1))))) -> 5^1(3(2(x1))) 34.24/9.58 The remaining pairs can at least be oriented weakly. 34.24/9.58 Used ordering: Polynomial interpretation [POLO]: 34.24/9.58 34.24/9.58 POL(0(x_1)) = 0 34.24/9.58 POL(1(x_1)) = 1 34.24/9.58 POL(2(x_1)) = 1 + x_1 34.24/9.58 POL(3(x_1)) = x_1 34.24/9.58 POL(4(x_1)) = x_1 34.24/9.58 POL(5(x_1)) = 0 34.24/9.58 POL(5^1(x_1)) = x_1 34.24/9.58 34.24/9.58 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.24/9.58 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (13) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (14) UsableRulesProof (EQUIVALENT) 34.24/9.58 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. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (15) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 34.24/9.58 R is empty. 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (16) QDPSizeChangeProof (EQUIVALENT) 34.24/9.58 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. 34.24/9.58 34.24/9.58 From the DPs we obtained the following set of size-change graphs: 34.24/9.58 *5^1(4(3(3(x1)))) -> 5^1(x1) 34.24/9.58 The graph contains the following edges 1 > 1 34.24/9.58 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (17) 34.24/9.58 YES 34.24/9.58 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (18) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 0^1(4(1(2(3(x1))))) -> 0^1(3(2(4(5(1(x1)))))) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 0(1(1(x1))) -> 0(2(1(1(x1)))) 34.24/9.58 0(1(1(x1))) -> 0(0(2(1(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 0(2(1(2(1(x1))))) 34.24/9.58 0(1(1(x1))) -> 2(1(0(2(1(x1))))) 34.24/9.58 3(0(1(x1))) -> 1(3(0(0(2(4(x1)))))) 34.24/9.58 3(5(1(x1))) -> 1(3(4(5(x1)))) 34.24/9.58 3(5(1(x1))) -> 2(4(5(3(1(x1))))) 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 0(1(0(1(x1)))) -> 1(1(0(0(2(4(x1)))))) 34.24/9.58 0(1(2(3(x1)))) -> 3(1(5(0(2(x1))))) 34.24/9.58 0(1(2(3(x1)))) -> 0(3(1(2(1(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 0(2(1(2(4(1(x1)))))) 34.24/9.58 0(1(4(1(x1)))) -> 4(0(0(2(1(1(x1)))))) 34.24/9.58 0(1(5(1(x1)))) -> 0(0(2(1(1(5(x1)))))) 34.24/9.58 0(4(5(1(x1)))) -> 0(1(3(4(5(x1))))) 34.24/9.58 3(2(0(1(x1)))) -> 1(3(0(2(4(5(x1)))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(3(4(5(1(x1))))) 34.24/9.58 3(5(1(1(x1)))) -> 1(5(3(1(2(x1))))) 34.24/9.58 3(5(1(3(x1)))) -> 3(5(3(1(2(x1))))) 34.24/9.58 3(5(4(1(x1)))) -> 4(1(3(4(5(x1))))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 5(2(0(1(x1)))) -> 5(3(1(0(2(4(x1)))))) 34.24/9.58 5(4(3(3(x1)))) -> 3(1(3(4(5(x1))))) 34.24/9.58 5(5(0(1(x1)))) -> 1(0(2(4(5(5(x1)))))) 34.24/9.58 0(1(2(0(1(x1))))) -> 0(3(0(2(1(1(x1)))))) 34.24/9.58 0(1(2(2(1(x1))))) -> 0(2(1(2(1(3(x1)))))) 34.24/9.58 0(3(0(5(1(x1))))) -> 0(3(4(5(0(1(x1)))))) 34.24/9.58 0(3(4(2(3(x1))))) -> 0(5(3(4(3(2(x1)))))) 34.24/9.58 0(3(5(4(1(x1))))) -> 0(5(2(4(3(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 0(3(2(4(5(1(x1)))))) 34.24/9.58 0(4(1(2(3(x1))))) -> 4(3(1(0(0(2(x1)))))) 34.24/9.58 0(4(5(5(1(x1))))) -> 2(4(5(5(0(1(x1)))))) 34.24/9.58 0(5(1(0(1(x1))))) -> 0(1(5(5(0(1(x1)))))) 34.24/9.58 0(5(3(2(1(x1))))) -> 0(0(2(5(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 0(2(3(4(3(1(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(1(3(3(0(2(x1)))))) 34.24/9.58 3(0(1(2(3(x1))))) -> 1(2(3(3(0(2(x1)))))) 34.24/9.58 3(0(4(1(1(x1))))) -> 0(0(1(3(4(1(x1)))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 3(3(4(1(1(x1))))) -> 1(3(4(5(3(1(x1)))))) 34.24/9.58 3(3(5(1(1(x1))))) -> 3(1(4(5(3(1(x1)))))) 34.24/9.58 3(5(4(1(3(x1))))) -> 1(4(5(3(1(3(x1)))))) 34.24/9.58 3(5(4(4(1(x1))))) -> 4(1(4(3(4(5(x1)))))) 34.24/9.58 5(2(4(2(3(x1))))) -> 3(2(4(5(3(2(x1)))))) 34.24/9.58 5(4(2(0(1(x1))))) -> 5(1(2(0(2(4(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (19) UsableRulesProof (EQUIVALENT) 34.24/9.58 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. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (20) 34.24/9.58 Obligation: 34.24/9.58 Q DP problem: 34.24/9.58 The TRS P consists of the following rules: 34.24/9.58 34.24/9.58 0^1(4(1(2(3(x1))))) -> 0^1(3(2(4(5(1(x1)))))) 34.24/9.58 34.24/9.58 The TRS R consists of the following rules: 34.24/9.58 34.24/9.58 5(1(3(x1))) -> 5(3(1(2(x1)))) 34.24/9.58 5(1(2(3(x1)))) -> 5(5(3(1(2(x1))))) 34.24/9.58 3(2(4(1(3(x1))))) -> 4(3(4(3(1(2(x1)))))) 34.24/9.58 34.24/9.58 Q is empty. 34.24/9.58 We have to consider all minimal (P,Q,R)-chains. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (21) DependencyGraphProof (EQUIVALENT) 34.24/9.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 34.24/9.58 ---------------------------------------- 34.24/9.58 34.24/9.58 (22) 34.24/9.58 TRUE 34.66/9.68 EOF