14.72/4.69 YES 15.54/4.90 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 15.54/4.90 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 15.54/4.90 15.54/4.90 15.54/4.90 Termination w.r.t. Q of the given QTRS could be proven: 15.54/4.90 15.54/4.90 (0) QTRS 15.54/4.90 (1) QTRS Reverse [EQUIVALENT, 0 ms] 15.54/4.90 (2) QTRS 15.54/4.90 (3) DependencyPairsProof [EQUIVALENT, 301 ms] 15.54/4.90 (4) QDP 15.54/4.90 (5) DependencyGraphProof [EQUIVALENT, 12 ms] 15.54/4.90 (6) AND 15.54/4.90 (7) QDP 15.54/4.90 (8) UsableRulesProof [EQUIVALENT, 8 ms] 15.54/4.90 (9) QDP 15.54/4.90 (10) DependencyGraphProof [EQUIVALENT, 0 ms] 15.54/4.90 (11) QDP 15.54/4.90 (12) UsableRulesProof [EQUIVALENT, 0 ms] 15.54/4.90 (13) QDP 15.54/4.90 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 15.54/4.90 (15) YES 15.54/4.90 (16) QDP 15.54/4.90 (17) UsableRulesProof [EQUIVALENT, 9 ms] 15.54/4.90 (18) QDP 15.54/4.90 (19) DependencyGraphProof [EQUIVALENT, 0 ms] 15.54/4.90 (20) TRUE 15.54/4.90 15.54/4.90 15.54/4.90 ---------------------------------------- 15.54/4.90 15.54/4.90 (0) 15.54/4.90 Obligation: 15.54/4.90 Q restricted rewrite system: 15.54/4.90 The TRS R consists of the following rules: 15.54/4.90 15.54/4.90 0(0(1(2(x1)))) -> 0(2(0(1(1(x1))))) 15.54/4.90 0(1(2(2(x1)))) -> 0(2(2(1(0(x1))))) 15.54/4.90 0(1(2(2(x1)))) -> 1(0(2(2(0(x1))))) 15.54/4.91 0(1(2(3(x1)))) -> 0(2(0(1(3(x1))))) 15.54/4.91 0(1(2(3(x1)))) -> 0(2(3(3(1(x1))))) 15.54/4.91 0(2(1(2(x1)))) -> 0(2(2(0(1(x1))))) 15.54/4.91 0(2(1(2(x1)))) -> 0(2(2(2(1(x1))))) 15.54/4.91 0(2(1(2(x1)))) -> 0(2(2(2(1(1(x1)))))) 15.54/4.91 0(3(2(2(x1)))) -> 3(4(0(2(2(x1))))) 15.54/4.91 0(3(2(2(x1)))) -> 0(2(2(2(2(3(x1)))))) 15.54/4.91 0(4(1(2(x1)))) -> 0(2(2(1(4(x1))))) 15.54/4.91 0(4(1(2(x1)))) -> 4(0(2(0(1(x1))))) 15.54/4.91 0(5(0(1(x1)))) -> 0(2(0(2(5(1(x1)))))) 15.54/4.91 0(5(0(5(x1)))) -> 0(2(0(5(5(x1))))) 15.54/4.91 0(5(2(1(x1)))) -> 0(2(2(5(1(x1))))) 15.54/4.91 0(5(2(5(x1)))) -> 0(2(2(5(5(x1))))) 15.54/4.91 0(5(4(2(x1)))) -> 4(0(2(2(0(5(x1)))))) 15.54/4.91 2(1(0(3(x1)))) -> 4(0(2(2(3(1(x1)))))) 15.54/4.91 2(1(0(4(x1)))) -> 1(4(0(2(2(2(x1)))))) 15.54/4.91 2(5(4(2(x1)))) -> 4(0(2(2(5(x1))))) 15.54/4.91 0(0(1(0(4(x1))))) -> 0(0(2(0(1(4(x1)))))) 15.54/4.91 0(0(5(4(2(x1))))) -> 0(4(0(0(2(5(x1)))))) 15.54/4.91 0(1(0(1(2(x1))))) -> 0(2(0(1(4(1(x1)))))) 15.54/4.91 0(1(2(0(3(x1))))) -> 0(2(0(4(1(3(x1)))))) 15.54/4.91 0(1(2(2(2(x1))))) -> 0(2(2(2(1(2(x1)))))) 15.54/4.91 0(1(2(3(2(x1))))) -> 1(3(4(0(2(2(x1)))))) 15.54/4.91 0(1(3(2(3(x1))))) -> 0(0(2(3(3(1(x1)))))) 15.54/4.91 0(1(3(4(2(x1))))) -> 0(2(3(4(1(1(x1)))))) 15.54/4.91 0(2(1(0(1(x1))))) -> 0(0(2(0(1(1(x1)))))) 15.54/4.91 0(2(1(2(2(x1))))) -> 0(2(0(2(2(1(x1)))))) 15.54/4.91 0(2(3(0(5(x1))))) -> 0(2(0(0(5(3(x1)))))) 15.54/4.91 0(3(0(1(3(x1))))) -> 0(0(4(3(1(3(x1)))))) 15.54/4.91 0(3(0(4(1(x1))))) -> 0(0(1(4(4(3(x1)))))) 15.54/4.91 0(3(2(0(4(x1))))) -> 4(0(0(2(3(4(x1)))))) 15.54/4.91 0(4(5(2(3(x1))))) -> 0(2(2(3(4(5(x1)))))) 15.54/4.91 0(5(0(0(3(x1))))) -> 0(2(0(3(0(5(x1)))))) 15.54/4.91 0(5(0(1(2(x1))))) -> 0(0(2(0(1(5(x1)))))) 15.54/4.91 0(5(1(4(2(x1))))) -> 0(2(0(1(4(5(x1)))))) 15.54/4.91 0(5(2(5(1(x1))))) -> 0(2(0(5(5(1(x1)))))) 15.54/4.91 2(1(0(0(4(x1))))) -> 1(4(4(0(0(2(x1)))))) 15.54/4.91 2(5(0(0(3(x1))))) -> 0(2(0(0(5(3(x1)))))) 15.54/4.91 2(5(3(0(1(x1))))) -> 5(0(2(2(3(1(x1)))))) 15.54/4.91 5(0(1(2(2(x1))))) -> 5(1(0(2(0(2(x1)))))) 15.54/4.91 5(2(0(1(2(x1))))) -> 1(5(4(0(2(2(x1)))))) 15.54/4.91 5(2(1(0(1(x1))))) -> 0(2(3(1(5(1(x1)))))) 15.54/4.91 5(2(3(0(1(x1))))) -> 1(5(0(2(2(3(x1)))))) 15.54/4.91 5(3(0(4(1(x1))))) -> 4(5(0(2(3(1(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (1) QTRS Reverse (EQUIVALENT) 15.54/4.91 We applied the QTRS Reverse Processor [REVERSE]. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (2) 15.54/4.91 Obligation: 15.54/4.91 Q restricted rewrite system: 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 2(1(0(0(x1)))) -> 1(1(0(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(1(2(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(2(2(0(1(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 3(1(0(2(0(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 1(3(3(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(0(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(2(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(1(2(2(2(0(x1)))))) 15.54/4.91 2(2(3(0(x1)))) -> 2(2(0(4(3(x1))))) 15.54/4.91 2(2(3(0(x1)))) -> 3(2(2(2(2(0(x1)))))) 15.54/4.91 2(1(4(0(x1)))) -> 4(1(2(2(0(x1))))) 15.54/4.91 2(1(4(0(x1)))) -> 1(0(2(0(4(x1))))) 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 2(4(5(0(x1)))) -> 5(0(2(2(0(4(x1)))))) 15.54/4.91 3(0(1(2(x1)))) -> 1(3(2(2(0(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 2(4(5(2(x1)))) -> 5(2(2(0(4(x1))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(5(0(0(x1))))) -> 5(2(0(0(4(0(x1)))))) 15.54/4.91 2(1(0(1(0(x1))))) -> 1(4(1(0(2(0(x1)))))) 15.54/4.91 3(0(2(1(0(x1))))) -> 3(1(4(0(2(0(x1)))))) 15.54/4.91 2(2(2(1(0(x1))))) -> 2(1(2(2(2(0(x1)))))) 15.54/4.91 2(3(2(1(0(x1))))) -> 2(2(0(4(3(1(x1)))))) 15.54/4.91 3(2(3(1(0(x1))))) -> 1(3(3(2(0(0(x1)))))) 15.54/4.91 2(4(3(1(0(x1))))) -> 1(1(4(3(2(0(x1)))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 2(2(1(2(0(x1))))) -> 1(2(2(0(2(0(x1)))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 3(1(0(3(0(x1))))) -> 3(1(3(4(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 3(2(5(4(0(x1))))) -> 5(4(3(2(2(0(x1)))))) 15.54/4.91 3(0(0(5(0(x1))))) -> 5(0(3(0(2(0(x1)))))) 15.54/4.91 2(1(0(5(0(x1))))) -> 5(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(1(5(0(x1))))) -> 5(4(1(0(2(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 3(0(0(5(2(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 2(2(1(0(5(x1))))) -> 2(0(2(0(1(5(x1)))))) 15.54/4.91 2(1(0(2(5(x1))))) -> 2(2(0(4(5(1(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (3) DependencyPairsProof (EQUIVALENT) 15.54/4.91 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (4) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 2^1(1(0(0(x1)))) -> 1^1(1(0(2(0(x1))))) 15.54/4.91 2^1(1(0(0(x1)))) -> 1^1(0(2(0(x1)))) 15.54/4.91 2^1(1(0(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 2^1(2(1(0(x1)))) -> 1^1(2(2(0(x1)))) 15.54/4.91 2^1(2(1(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 2^1(2(1(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 2^1(2(1(0(x1)))) -> 2^1(2(0(1(x1)))) 15.54/4.91 2^1(2(1(0(x1)))) -> 2^1(0(1(x1))) 15.54/4.91 2^1(2(1(0(x1)))) -> 1^1(x1) 15.54/4.91 3^1(2(1(0(x1)))) -> 3^1(1(0(2(0(x1))))) 15.54/4.91 3^1(2(1(0(x1)))) -> 1^1(0(2(0(x1)))) 15.54/4.91 3^1(2(1(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 3^1(2(1(0(x1)))) -> 1^1(3(3(2(0(x1))))) 15.54/4.91 3^1(2(1(0(x1)))) -> 3^1(3(2(0(x1)))) 15.54/4.91 3^1(2(1(0(x1)))) -> 3^1(2(0(x1))) 15.54/4.91 2^1(1(2(0(x1)))) -> 1^1(0(2(2(0(x1))))) 15.54/4.91 2^1(1(2(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 2^1(1(2(0(x1)))) -> 1^1(2(2(2(0(x1))))) 15.54/4.91 2^1(1(2(0(x1)))) -> 2^1(2(2(0(x1)))) 15.54/4.91 2^1(1(2(0(x1)))) -> 1^1(1(2(2(2(0(x1)))))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(2(0(4(3(x1))))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(0(4(3(x1)))) 15.54/4.91 2^1(2(3(0(x1)))) -> 4^1(3(x1)) 15.54/4.91 2^1(2(3(0(x1)))) -> 3^1(x1) 15.54/4.91 2^1(2(3(0(x1)))) -> 3^1(2(2(2(2(0(x1)))))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(2(2(2(0(x1))))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(2(2(0(x1)))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 2^1(2(3(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 2^1(1(4(0(x1)))) -> 4^1(1(2(2(0(x1))))) 15.54/4.91 2^1(1(4(0(x1)))) -> 1^1(2(2(0(x1)))) 15.54/4.91 2^1(1(4(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 2^1(1(4(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 2^1(1(4(0(x1)))) -> 1^1(0(2(0(4(x1))))) 15.54/4.91 2^1(1(4(0(x1)))) -> 2^1(0(4(x1))) 15.54/4.91 2^1(1(4(0(x1)))) -> 4^1(x1) 15.54/4.91 1^1(0(5(0(x1)))) -> 1^1(5(2(0(2(0(x1)))))) 15.54/4.91 1^1(0(5(0(x1)))) -> 5^1(2(0(2(0(x1))))) 15.54/4.91 1^1(0(5(0(x1)))) -> 2^1(0(2(0(x1)))) 15.54/4.91 1^1(0(5(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 5^1(0(5(0(x1)))) -> 5^1(5(0(2(0(x1))))) 15.54/4.91 5^1(0(5(0(x1)))) -> 5^1(0(2(0(x1)))) 15.54/4.91 5^1(0(5(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 1^1(2(5(0(x1)))) -> 1^1(5(2(2(0(x1))))) 15.54/4.91 1^1(2(5(0(x1)))) -> 5^1(2(2(0(x1)))) 15.54/4.91 1^1(2(5(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 1^1(2(5(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 5^1(2(5(0(x1)))) -> 5^1(5(2(2(0(x1))))) 15.54/4.91 5^1(2(5(0(x1)))) -> 5^1(2(2(0(x1)))) 15.54/4.91 5^1(2(5(0(x1)))) -> 2^1(2(0(x1))) 15.54/4.91 5^1(2(5(0(x1)))) -> 2^1(0(x1)) 15.54/4.91 2^1(4(5(0(x1)))) -> 5^1(0(2(2(0(4(x1)))))) 15.54/4.91 2^1(4(5(0(x1)))) -> 2^1(2(0(4(x1)))) 15.54/4.91 2^1(4(5(0(x1)))) -> 2^1(0(4(x1))) 15.54/4.91 2^1(4(5(0(x1)))) -> 4^1(x1) 15.54/4.91 3^1(0(1(2(x1)))) -> 1^1(3(2(2(0(4(x1)))))) 15.54/4.91 3^1(0(1(2(x1)))) -> 3^1(2(2(0(4(x1))))) 15.54/4.91 3^1(0(1(2(x1)))) -> 2^1(2(0(4(x1)))) 15.54/4.91 3^1(0(1(2(x1)))) -> 2^1(0(4(x1))) 15.54/4.91 3^1(0(1(2(x1)))) -> 4^1(x1) 15.54/4.91 4^1(0(1(2(x1)))) -> 2^1(2(2(0(4(1(x1)))))) 15.54/4.91 4^1(0(1(2(x1)))) -> 2^1(2(0(4(1(x1))))) 15.54/4.91 4^1(0(1(2(x1)))) -> 2^1(0(4(1(x1)))) 15.54/4.91 4^1(0(1(2(x1)))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 2^1(4(5(2(x1)))) -> 5^1(2(2(0(4(x1))))) 15.54/4.91 2^1(4(5(2(x1)))) -> 2^1(2(0(4(x1)))) 15.54/4.91 2^1(4(5(2(x1)))) -> 2^1(0(4(x1))) 15.54/4.91 2^1(4(5(2(x1)))) -> 4^1(x1) 15.54/4.91 4^1(0(1(0(0(x1))))) -> 4^1(1(0(2(0(0(x1)))))) 15.54/4.91 4^1(0(1(0(0(x1))))) -> 1^1(0(2(0(0(x1))))) 15.54/4.91 4^1(0(1(0(0(x1))))) -> 2^1(0(0(x1))) 15.54/4.91 2^1(4(5(0(0(x1))))) -> 5^1(2(0(0(4(0(x1)))))) 15.54/4.91 2^1(4(5(0(0(x1))))) -> 2^1(0(0(4(0(x1))))) 15.54/4.91 2^1(4(5(0(0(x1))))) -> 4^1(0(x1)) 15.54/4.91 2^1(1(0(1(0(x1))))) -> 1^1(4(1(0(2(0(x1)))))) 15.54/4.91 2^1(1(0(1(0(x1))))) -> 4^1(1(0(2(0(x1))))) 15.54/4.91 2^1(1(0(1(0(x1))))) -> 1^1(0(2(0(x1)))) 15.54/4.91 2^1(1(0(1(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 3^1(0(2(1(0(x1))))) -> 3^1(1(4(0(2(0(x1)))))) 15.54/4.91 3^1(0(2(1(0(x1))))) -> 1^1(4(0(2(0(x1))))) 15.54/4.91 3^1(0(2(1(0(x1))))) -> 4^1(0(2(0(x1)))) 15.54/4.91 3^1(0(2(1(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 2^1(2(2(1(0(x1))))) -> 2^1(1(2(2(2(0(x1)))))) 15.54/4.91 2^1(2(2(1(0(x1))))) -> 1^1(2(2(2(0(x1))))) 15.54/4.91 2^1(2(2(1(0(x1))))) -> 2^1(2(2(0(x1)))) 15.54/4.91 2^1(2(2(1(0(x1))))) -> 2^1(2(0(x1))) 15.54/4.91 2^1(2(2(1(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 2^1(3(2(1(0(x1))))) -> 2^1(2(0(4(3(1(x1)))))) 15.54/4.91 2^1(3(2(1(0(x1))))) -> 2^1(0(4(3(1(x1))))) 15.54/4.91 2^1(3(2(1(0(x1))))) -> 4^1(3(1(x1))) 15.54/4.91 2^1(3(2(1(0(x1))))) -> 3^1(1(x1)) 15.54/4.91 2^1(3(2(1(0(x1))))) -> 1^1(x1) 15.54/4.91 3^1(2(3(1(0(x1))))) -> 1^1(3(3(2(0(0(x1)))))) 15.54/4.91 3^1(2(3(1(0(x1))))) -> 3^1(3(2(0(0(x1))))) 15.54/4.91 3^1(2(3(1(0(x1))))) -> 3^1(2(0(0(x1)))) 15.54/4.91 3^1(2(3(1(0(x1))))) -> 2^1(0(0(x1))) 15.54/4.91 2^1(4(3(1(0(x1))))) -> 1^1(1(4(3(2(0(x1)))))) 15.54/4.91 2^1(4(3(1(0(x1))))) -> 1^1(4(3(2(0(x1))))) 15.54/4.91 2^1(4(3(1(0(x1))))) -> 4^1(3(2(0(x1)))) 15.54/4.91 2^1(4(3(1(0(x1))))) -> 3^1(2(0(x1))) 15.54/4.91 2^1(4(3(1(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 1^1(0(1(2(0(x1))))) -> 1^1(1(0(2(0(0(x1)))))) 15.54/4.91 1^1(0(1(2(0(x1))))) -> 1^1(0(2(0(0(x1))))) 15.54/4.91 1^1(0(1(2(0(x1))))) -> 2^1(0(0(x1))) 15.54/4.91 2^1(2(1(2(0(x1))))) -> 1^1(2(2(0(2(0(x1)))))) 15.54/4.91 2^1(2(1(2(0(x1))))) -> 2^1(2(0(2(0(x1))))) 15.54/4.91 2^1(2(1(2(0(x1))))) -> 2^1(0(2(0(x1)))) 15.54/4.91 5^1(0(3(2(0(x1))))) -> 3^1(5(0(0(2(0(x1)))))) 15.54/4.91 5^1(0(3(2(0(x1))))) -> 5^1(0(0(2(0(x1))))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(1(3(4(0(0(x1)))))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 1^1(3(4(0(0(x1))))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(4(0(0(x1)))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 4^1(0(0(x1))) 15.54/4.91 1^1(4(0(3(0(x1))))) -> 3^1(4(4(1(0(0(x1)))))) 15.54/4.91 1^1(4(0(3(0(x1))))) -> 4^1(4(1(0(0(x1))))) 15.54/4.91 1^1(4(0(3(0(x1))))) -> 4^1(1(0(0(x1)))) 15.54/4.91 1^1(4(0(3(0(x1))))) -> 1^1(0(0(x1))) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(3(2(0(0(4(x1)))))) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 3^1(2(0(0(4(x1))))) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 2^1(0(0(4(x1)))) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 3^1(2(5(4(0(x1))))) -> 5^1(4(3(2(2(0(x1)))))) 15.54/4.91 3^1(2(5(4(0(x1))))) -> 4^1(3(2(2(0(x1))))) 15.54/4.91 3^1(2(5(4(0(x1))))) -> 3^1(2(2(0(x1)))) 15.54/4.91 3^1(2(5(4(0(x1))))) -> 2^1(2(0(x1))) 15.54/4.91 3^1(2(5(4(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 3^1(0(0(5(0(x1))))) -> 5^1(0(3(0(2(0(x1)))))) 15.54/4.91 3^1(0(0(5(0(x1))))) -> 3^1(0(2(0(x1)))) 15.54/4.91 3^1(0(0(5(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 2^1(1(0(5(0(x1))))) -> 5^1(1(0(2(0(0(x1)))))) 15.54/4.91 2^1(1(0(5(0(x1))))) -> 1^1(0(2(0(0(x1))))) 15.54/4.91 2^1(1(0(5(0(x1))))) -> 2^1(0(0(x1))) 15.54/4.91 2^1(4(1(5(0(x1))))) -> 5^1(4(1(0(2(0(x1)))))) 15.54/4.91 2^1(4(1(5(0(x1))))) -> 4^1(1(0(2(0(x1))))) 15.54/4.91 2^1(4(1(5(0(x1))))) -> 1^1(0(2(0(x1)))) 15.54/4.91 2^1(4(1(5(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 1^1(5(2(5(0(x1))))) -> 1^1(5(5(0(2(0(x1)))))) 15.54/4.91 1^1(5(2(5(0(x1))))) -> 5^1(5(0(2(0(x1))))) 15.54/4.91 1^1(5(2(5(0(x1))))) -> 5^1(0(2(0(x1)))) 15.54/4.91 1^1(5(2(5(0(x1))))) -> 2^1(0(x1)) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 2^1(0(0(4(4(1(x1)))))) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(4(1(x1))) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 3^1(0(0(5(2(x1))))) -> 3^1(5(0(0(2(0(x1)))))) 15.54/4.91 3^1(0(0(5(2(x1))))) -> 5^1(0(0(2(0(x1))))) 15.54/4.91 3^1(0(0(5(2(x1))))) -> 2^1(0(x1)) 15.54/4.91 1^1(0(3(5(2(x1))))) -> 1^1(3(2(2(0(5(x1)))))) 15.54/4.91 1^1(0(3(5(2(x1))))) -> 3^1(2(2(0(5(x1))))) 15.54/4.91 1^1(0(3(5(2(x1))))) -> 2^1(2(0(5(x1)))) 15.54/4.91 1^1(0(3(5(2(x1))))) -> 2^1(0(5(x1))) 15.54/4.91 1^1(0(3(5(2(x1))))) -> 5^1(x1) 15.54/4.91 2^1(2(1(0(5(x1))))) -> 2^1(0(2(0(1(5(x1)))))) 15.54/4.91 2^1(2(1(0(5(x1))))) -> 2^1(0(1(5(x1)))) 15.54/4.91 2^1(2(1(0(5(x1))))) -> 1^1(5(x1)) 15.54/4.91 2^1(1(0(2(5(x1))))) -> 2^1(2(0(4(5(1(x1)))))) 15.54/4.91 2^1(1(0(2(5(x1))))) -> 2^1(0(4(5(1(x1))))) 15.54/4.91 2^1(1(0(2(5(x1))))) -> 4^1(5(1(x1))) 15.54/4.91 2^1(1(0(2(5(x1))))) -> 5^1(1(x1)) 15.54/4.91 2^1(1(0(2(5(x1))))) -> 1^1(x1) 15.54/4.91 1^1(0(1(2(5(x1))))) -> 1^1(5(1(3(2(0(x1)))))) 15.54/4.91 1^1(0(1(2(5(x1))))) -> 5^1(1(3(2(0(x1))))) 15.54/4.91 1^1(0(1(2(5(x1))))) -> 1^1(3(2(0(x1)))) 15.54/4.91 1^1(0(1(2(5(x1))))) -> 3^1(2(0(x1))) 15.54/4.91 1^1(0(1(2(5(x1))))) -> 2^1(0(x1)) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 3^1(2(2(0(5(1(x1)))))) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 2^1(2(0(5(1(x1))))) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 2^1(0(5(1(x1)))) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 5^1(1(x1)) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 1^1(3(2(0(5(4(x1)))))) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 3^1(2(0(5(4(x1))))) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 2^1(0(5(4(x1)))) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 5^1(4(x1)) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 2(1(0(0(x1)))) -> 1(1(0(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(1(2(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(2(2(0(1(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 3(1(0(2(0(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 1(3(3(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(0(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(2(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(1(2(2(2(0(x1)))))) 15.54/4.91 2(2(3(0(x1)))) -> 2(2(0(4(3(x1))))) 15.54/4.91 2(2(3(0(x1)))) -> 3(2(2(2(2(0(x1)))))) 15.54/4.91 2(1(4(0(x1)))) -> 4(1(2(2(0(x1))))) 15.54/4.91 2(1(4(0(x1)))) -> 1(0(2(0(4(x1))))) 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 2(4(5(0(x1)))) -> 5(0(2(2(0(4(x1)))))) 15.54/4.91 3(0(1(2(x1)))) -> 1(3(2(2(0(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 2(4(5(2(x1)))) -> 5(2(2(0(4(x1))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(5(0(0(x1))))) -> 5(2(0(0(4(0(x1)))))) 15.54/4.91 2(1(0(1(0(x1))))) -> 1(4(1(0(2(0(x1)))))) 15.54/4.91 3(0(2(1(0(x1))))) -> 3(1(4(0(2(0(x1)))))) 15.54/4.91 2(2(2(1(0(x1))))) -> 2(1(2(2(2(0(x1)))))) 15.54/4.91 2(3(2(1(0(x1))))) -> 2(2(0(4(3(1(x1)))))) 15.54/4.91 3(2(3(1(0(x1))))) -> 1(3(3(2(0(0(x1)))))) 15.54/4.91 2(4(3(1(0(x1))))) -> 1(1(4(3(2(0(x1)))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 2(2(1(2(0(x1))))) -> 1(2(2(0(2(0(x1)))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 3(1(0(3(0(x1))))) -> 3(1(3(4(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 3(2(5(4(0(x1))))) -> 5(4(3(2(2(0(x1)))))) 15.54/4.91 3(0(0(5(0(x1))))) -> 5(0(3(0(2(0(x1)))))) 15.54/4.91 2(1(0(5(0(x1))))) -> 5(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(1(5(0(x1))))) -> 5(4(1(0(2(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 3(0(0(5(2(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 2(2(1(0(5(x1))))) -> 2(0(2(0(1(5(x1)))))) 15.54/4.91 2(1(0(2(5(x1))))) -> 2(2(0(4(5(1(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (5) DependencyGraphProof (EQUIVALENT) 15.54/4.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 166 less nodes. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (6) 15.54/4.91 Complex Obligation (AND) 15.54/4.91 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (7) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(1(2(x1)))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(4(1(x1))) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 2(1(0(0(x1)))) -> 1(1(0(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(1(2(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(2(2(0(1(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 3(1(0(2(0(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 1(3(3(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(0(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(2(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(1(2(2(2(0(x1)))))) 15.54/4.91 2(2(3(0(x1)))) -> 2(2(0(4(3(x1))))) 15.54/4.91 2(2(3(0(x1)))) -> 3(2(2(2(2(0(x1)))))) 15.54/4.91 2(1(4(0(x1)))) -> 4(1(2(2(0(x1))))) 15.54/4.91 2(1(4(0(x1)))) -> 1(0(2(0(4(x1))))) 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 2(4(5(0(x1)))) -> 5(0(2(2(0(4(x1)))))) 15.54/4.91 3(0(1(2(x1)))) -> 1(3(2(2(0(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 2(4(5(2(x1)))) -> 5(2(2(0(4(x1))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(5(0(0(x1))))) -> 5(2(0(0(4(0(x1)))))) 15.54/4.91 2(1(0(1(0(x1))))) -> 1(4(1(0(2(0(x1)))))) 15.54/4.91 3(0(2(1(0(x1))))) -> 3(1(4(0(2(0(x1)))))) 15.54/4.91 2(2(2(1(0(x1))))) -> 2(1(2(2(2(0(x1)))))) 15.54/4.91 2(3(2(1(0(x1))))) -> 2(2(0(4(3(1(x1)))))) 15.54/4.91 3(2(3(1(0(x1))))) -> 1(3(3(2(0(0(x1)))))) 15.54/4.91 2(4(3(1(0(x1))))) -> 1(1(4(3(2(0(x1)))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 2(2(1(2(0(x1))))) -> 1(2(2(0(2(0(x1)))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 3(1(0(3(0(x1))))) -> 3(1(3(4(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 3(2(5(4(0(x1))))) -> 5(4(3(2(2(0(x1)))))) 15.54/4.91 3(0(0(5(0(x1))))) -> 5(0(3(0(2(0(x1)))))) 15.54/4.91 2(1(0(5(0(x1))))) -> 5(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(1(5(0(x1))))) -> 5(4(1(0(2(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 3(0(0(5(2(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 2(2(1(0(5(x1))))) -> 2(0(2(0(1(5(x1)))))) 15.54/4.91 2(1(0(2(5(x1))))) -> 2(2(0(4(5(1(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (8) UsableRulesProof (EQUIVALENT) 15.54/4.91 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. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (9) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(1(2(x1)))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(4(1(x1))) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 4^1(1(x1)) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (10) DependencyGraphProof (EQUIVALENT) 15.54/4.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (11) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (12) UsableRulesProof (EQUIVALENT) 15.54/4.91 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. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (13) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 15.54/4.91 R is empty. 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (14) QDPSizeChangeProof (EQUIVALENT) 15.54/4.91 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. 15.54/4.91 15.54/4.91 From the DPs we obtained the following set of size-change graphs: 15.54/4.91 *4^1(0(2(3(0(x1))))) -> 4^1(x1) 15.54/4.91 The graph contains the following edges 1 > 1 15.54/4.91 15.54/4.91 15.54/4.91 *1^1(0(3(2(5(x1))))) -> 1^1(x1) 15.54/4.91 The graph contains the following edges 1 > 1 15.54/4.91 15.54/4.91 15.54/4.91 *1^1(4(0(3(5(x1))))) -> 4^1(x1) 15.54/4.91 The graph contains the following edges 1 > 1 15.54/4.91 15.54/4.91 15.54/4.91 *4^1(0(1(2(x1)))) -> 1^1(x1) 15.54/4.91 The graph contains the following edges 1 > 1 15.54/4.91 15.54/4.91 15.54/4.91 *4^1(0(0(1(2(x1))))) -> 1^1(x1) 15.54/4.91 The graph contains the following edges 1 > 1 15.54/4.91 15.54/4.91 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (15) 15.54/4.91 YES 15.54/4.91 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (16) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(4(0(0(x1)))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(1(3(4(0(0(x1)))))) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 2(1(0(0(x1)))) -> 1(1(0(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(1(2(2(0(x1))))) 15.54/4.91 2(2(1(0(x1)))) -> 0(2(2(0(1(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 3(1(0(2(0(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 1(3(3(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(0(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(2(2(2(0(x1))))) 15.54/4.91 2(1(2(0(x1)))) -> 1(1(2(2(2(0(x1)))))) 15.54/4.91 2(2(3(0(x1)))) -> 2(2(0(4(3(x1))))) 15.54/4.91 2(2(3(0(x1)))) -> 3(2(2(2(2(0(x1)))))) 15.54/4.91 2(1(4(0(x1)))) -> 4(1(2(2(0(x1))))) 15.54/4.91 2(1(4(0(x1)))) -> 1(0(2(0(4(x1))))) 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 2(4(5(0(x1)))) -> 5(0(2(2(0(4(x1)))))) 15.54/4.91 3(0(1(2(x1)))) -> 1(3(2(2(0(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 2(4(5(2(x1)))) -> 5(2(2(0(4(x1))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(5(0(0(x1))))) -> 5(2(0(0(4(0(x1)))))) 15.54/4.91 2(1(0(1(0(x1))))) -> 1(4(1(0(2(0(x1)))))) 15.54/4.91 3(0(2(1(0(x1))))) -> 3(1(4(0(2(0(x1)))))) 15.54/4.91 2(2(2(1(0(x1))))) -> 2(1(2(2(2(0(x1)))))) 15.54/4.91 2(3(2(1(0(x1))))) -> 2(2(0(4(3(1(x1)))))) 15.54/4.91 3(2(3(1(0(x1))))) -> 1(3(3(2(0(0(x1)))))) 15.54/4.91 2(4(3(1(0(x1))))) -> 1(1(4(3(2(0(x1)))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 2(2(1(2(0(x1))))) -> 1(2(2(0(2(0(x1)))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 3(1(0(3(0(x1))))) -> 3(1(3(4(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 3(2(5(4(0(x1))))) -> 5(4(3(2(2(0(x1)))))) 15.54/4.91 3(0(0(5(0(x1))))) -> 5(0(3(0(2(0(x1)))))) 15.54/4.91 2(1(0(5(0(x1))))) -> 5(1(0(2(0(0(x1)))))) 15.54/4.91 2(4(1(5(0(x1))))) -> 5(4(1(0(2(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 3(0(0(5(2(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 2(2(1(0(5(x1))))) -> 2(0(2(0(1(5(x1)))))) 15.54/4.91 2(1(0(2(5(x1))))) -> 2(2(0(4(5(1(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (17) UsableRulesProof (EQUIVALENT) 15.54/4.91 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. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (18) 15.54/4.91 Obligation: 15.54/4.91 Q DP problem: 15.54/4.91 The TRS P consists of the following rules: 15.54/4.91 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(4(0(0(x1)))) 15.54/4.91 3^1(1(0(3(0(x1))))) -> 3^1(1(3(4(0(0(x1)))))) 15.54/4.91 15.54/4.91 The TRS R consists of the following rules: 15.54/4.91 15.54/4.91 4(0(0(1(2(x1))))) -> 2(0(0(4(4(1(x1)))))) 15.54/4.91 3(2(1(0(x1)))) -> 3(1(0(2(0(x1))))) 15.54/4.91 3(2(1(0(x1)))) -> 1(3(3(2(0(x1))))) 15.54/4.91 3(0(1(2(x1)))) -> 1(3(2(2(0(4(x1)))))) 15.54/4.91 3(0(2(1(0(x1))))) -> 3(1(4(0(2(0(x1)))))) 15.54/4.91 3(2(3(1(0(x1))))) -> 1(3(3(2(0(0(x1)))))) 15.54/4.91 3(1(0(3(0(x1))))) -> 3(1(3(4(0(0(x1)))))) 15.54/4.91 3(2(5(4(0(x1))))) -> 5(4(3(2(2(0(x1)))))) 15.54/4.91 3(0(0(5(0(x1))))) -> 5(0(3(0(2(0(x1)))))) 15.54/4.91 3(0(0(5(2(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 1(0(5(0(x1)))) -> 1(5(2(0(2(0(x1)))))) 15.54/4.91 1(2(5(0(x1)))) -> 1(5(2(2(0(x1))))) 15.54/4.91 1(0(1(2(0(x1))))) -> 1(1(0(2(0(0(x1)))))) 15.54/4.91 1(4(0(3(0(x1))))) -> 3(4(4(1(0(0(x1)))))) 15.54/4.91 1(5(2(5(0(x1))))) -> 1(5(5(0(2(0(x1)))))) 15.54/4.91 1(0(3(5(2(x1))))) -> 1(3(2(2(0(5(x1)))))) 15.54/4.91 1(0(1(2(5(x1))))) -> 1(5(1(3(2(0(x1)))))) 15.54/4.91 1(0(3(2(5(x1))))) -> 3(2(2(0(5(1(x1)))))) 15.54/4.91 1(4(0(3(5(x1))))) -> 1(3(2(0(5(4(x1)))))) 15.54/4.91 4(0(1(2(x1)))) -> 2(2(2(0(4(1(x1)))))) 15.54/4.91 4(0(1(0(0(x1))))) -> 4(1(0(2(0(0(x1)))))) 15.54/4.91 4(0(2(3(0(x1))))) -> 4(3(2(0(0(4(x1)))))) 15.54/4.91 5(0(5(0(x1)))) -> 5(5(0(2(0(x1))))) 15.54/4.91 5(2(5(0(x1)))) -> 5(5(2(2(0(x1))))) 15.54/4.91 5(0(3(2(0(x1))))) -> 3(5(0(0(2(0(x1)))))) 15.54/4.91 15.54/4.91 Q is empty. 15.54/4.91 We have to consider all minimal (P,Q,R)-chains. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (19) DependencyGraphProof (EQUIVALENT) 15.54/4.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 15.54/4.91 ---------------------------------------- 15.54/4.91 15.54/4.91 (20) 15.54/4.91 TRUE 15.85/5.00 EOF