37.81/10.54 YES 37.81/10.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 37.81/10.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 37.81/10.57 37.81/10.57 37.81/10.57 Termination w.r.t. Q of the given QTRS could be proven: 37.81/10.57 37.81/10.57 (0) QTRS 37.81/10.57 (1) QTRS Reverse [EQUIVALENT, 0 ms] 37.81/10.57 (2) QTRS 37.81/10.57 (3) DependencyPairsProof [EQUIVALENT, 264 ms] 37.81/10.57 (4) QDP 37.81/10.57 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 37.81/10.57 (6) AND 37.81/10.57 (7) QDP 37.81/10.57 (8) UsableRulesProof [EQUIVALENT, 0 ms] 37.81/10.57 (9) QDP 37.81/10.57 (10) QDPOrderProof [EQUIVALENT, 11 ms] 37.81/10.57 (11) QDP 37.81/10.57 (12) PisEmptyProof [EQUIVALENT, 0 ms] 37.81/10.57 (13) YES 37.81/10.57 (14) QDP 37.81/10.57 (15) UsableRulesProof [EQUIVALENT, 0 ms] 37.81/10.57 (16) QDP 37.81/10.57 (17) QDPOrderProof [EQUIVALENT, 164 ms] 37.81/10.57 (18) QDP 37.81/10.57 (19) UsableRulesProof [EQUIVALENT, 0 ms] 37.81/10.57 (20) QDP 37.81/10.57 (21) MRRProof [EQUIVALENT, 0 ms] 37.81/10.57 (22) QDP 37.81/10.57 (23) PisEmptyProof [EQUIVALENT, 0 ms] 37.81/10.57 (24) YES 37.81/10.57 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (0) 37.81/10.57 Obligation: 37.81/10.57 Q restricted rewrite system: 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 0(1(1(x1))) -> 1(2(1(2(0(x1))))) 37.81/10.57 0(3(1(x1))) -> 1(3(2(2(0(x1))))) 37.81/10.57 0(3(1(x1))) -> 3(2(1(2(0(x1))))) 37.81/10.57 0(3(1(x1))) -> 1(3(3(3(2(0(x1)))))) 37.81/10.57 0(4(1(x1))) -> 2(1(2(0(4(x1))))) 37.81/10.57 0(0(4(5(x1)))) -> 0(0(2(5(4(x1))))) 37.81/10.57 0(1(4(1(x1)))) -> 0(1(2(2(4(1(x1)))))) 37.81/10.57 0(1(4(5(x1)))) -> 4(0(1(2(5(4(x1)))))) 37.81/10.57 0(1(5(1(x1)))) -> 1(2(2(5(0(1(x1)))))) 37.81/10.57 0(1(5(3(x1)))) -> 0(5(3(2(1(x1))))) 37.81/10.57 0(2(4(1(x1)))) -> 1(3(3(2(0(4(x1)))))) 37.81/10.57 0(2(4(1(x1)))) -> 4(2(1(2(0(4(x1)))))) 37.81/10.57 0(2(4(5(x1)))) -> 0(2(2(5(0(4(x1)))))) 37.81/10.57 0(3(1(5(x1)))) -> 0(1(2(5(3(x1))))) 37.81/10.57 0(3(1(5(x1)))) -> 1(2(5(3(0(4(x1)))))) 37.81/10.57 0(3(5(1(x1)))) -> 1(2(5(3(0(x1))))) 37.81/10.57 0(3(5(1(x1)))) -> 0(5(2(1(2(3(x1)))))) 37.81/10.57 0(3(5(5(x1)))) -> 0(3(2(5(5(x1))))) 37.81/10.57 0(4(0(1(x1)))) -> 2(0(4(4(0(1(x1)))))) 37.81/10.57 0(4(1(5(x1)))) -> 1(2(5(0(4(x1))))) 37.81/10.57 0(4(3(5(x1)))) -> 0(4(3(2(5(4(x1)))))) 37.81/10.57 0(4(5(1(x1)))) -> 2(5(4(4(0(1(x1)))))) 37.81/10.57 3(0(1(5(x1)))) -> 3(1(4(0(5(4(x1)))))) 37.81/10.57 3(0(3(1(x1)))) -> 1(3(3(2(0(x1))))) 37.81/10.57 3(0(3(5(x1)))) -> 3(2(5(0(2(3(x1)))))) 37.81/10.57 3(3(0(1(x1)))) -> 0(1(3(2(2(3(x1)))))) 37.81/10.57 3(4(5(1(x1)))) -> 3(2(5(4(2(1(x1)))))) 37.81/10.57 4(1(3(5(x1)))) -> 1(2(5(3(4(4(x1)))))) 37.81/10.57 4(1(5(1(x1)))) -> 4(4(5(1(2(1(x1)))))) 37.81/10.57 4(4(1(5(x1)))) -> 4(1(2(5(4(x1))))) 37.81/10.57 0(1(4(5(5(x1))))) -> 0(5(1(4(2(5(x1)))))) 37.81/10.57 0(2(1(4(5(x1))))) -> 0(0(1(2(5(4(x1)))))) 37.81/10.57 0(2(1(5(5(x1))))) -> 0(1(2(2(5(5(x1)))))) 37.81/10.57 0(4(2(4(1(x1))))) -> 1(3(2(0(4(4(x1)))))) 37.81/10.57 0(4(5(4(3(x1))))) -> 2(5(0(4(4(3(x1)))))) 37.81/10.57 0(5(1(5(1(x1))))) -> 0(5(1(1(2(5(x1)))))) 37.81/10.57 0(5(2(1(5(x1))))) -> 1(2(5(5(0(4(x1)))))) 37.81/10.57 0(5(2(4(1(x1))))) -> 4(5(2(1(2(0(x1)))))) 37.81/10.57 3(0(1(4(1(x1))))) -> 0(4(4(1(3(1(x1)))))) 37.81/10.57 3(0(1(4(1(x1))))) -> 4(3(2(0(1(1(x1)))))) 37.81/10.57 3(0(3(5(5(x1))))) -> 3(3(2(5(0(5(x1)))))) 37.81/10.57 3(0(5(3(1(x1))))) -> 1(0(3(3(2(5(x1)))))) 37.81/10.57 4(0(1(4(1(x1))))) -> 4(4(0(1(3(1(x1)))))) 37.81/10.57 4(0(1(5(1(x1))))) -> 0(1(2(5(4(1(x1)))))) 37.81/10.57 4(0(2(4(5(x1))))) -> 4(0(2(5(0(4(x1)))))) 37.81/10.57 4(1(1(5(1(x1))))) -> 1(1(2(5(4(1(x1)))))) 37.81/10.57 4(5(1(4(1(x1))))) -> 4(4(1(2(1(5(x1)))))) 37.81/10.57 4(5(2(3(1(x1))))) -> 4(3(1(2(2(5(x1)))))) 37.81/10.57 4(5(4(3(1(x1))))) -> 4(1(2(5(3(4(x1)))))) 37.81/10.57 4(5(5(3(1(x1))))) -> 1(3(2(5(5(4(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (1) QTRS Reverse (EQUIVALENT) 37.81/10.57 We applied the QTRS Reverse Processor [REVERSE]. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (2) 37.81/10.57 Obligation: 37.81/10.57 Q restricted rewrite system: 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(1(3(0(x1)))) -> 3(5(2(1(0(x1))))) 37.81/10.57 5(1(3(0(x1)))) -> 4(0(3(5(2(1(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 5(5(3(0(x1)))) -> 5(5(2(3(0(x1))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 5(1(4(0(x1)))) -> 4(0(5(2(1(x1))))) 37.81/10.57 5(3(4(0(x1)))) -> 4(5(2(3(4(0(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 5(1(0(3(x1)))) -> 4(5(0(4(1(3(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 5(3(0(3(x1)))) -> 3(2(0(5(2(3(x1)))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 5(3(1(4(x1)))) -> 4(4(3(5(2(1(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 5(1(4(4(x1)))) -> 4(5(2(1(4(x1))))) 37.81/10.57 5(5(4(1(0(x1))))) -> 5(2(4(1(5(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(5(1(2(0(x1))))) -> 5(5(2(2(1(0(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 5(1(2(5(0(x1))))) -> 4(0(5(5(2(1(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 5(5(3(0(3(x1))))) -> 5(0(5(2(3(3(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (3) DependencyPairsProof (EQUIVALENT) 37.81/10.57 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (4) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(1(0(x1))) -> 1^1(2(1(x1))) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(3(0(x1))) -> 3^1(1(x1)) 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(3(0(x1))) -> 1^1(2(3(x1))) 37.81/10.57 1^1(3(0(x1))) -> 3^1(x1) 37.81/10.57 1^1(3(0(x1))) -> 3^1(3(3(1(x1)))) 37.81/10.57 1^1(3(0(x1))) -> 3^1(3(1(x1))) 37.81/10.57 1^1(4(0(x1))) -> 1^1(2(x1)) 37.81/10.57 5^1(4(0(0(x1)))) -> 5^1(2(0(0(x1)))) 37.81/10.57 1^1(4(1(0(x1)))) -> 1^1(4(2(2(1(0(x1)))))) 37.81/10.57 5^1(4(1(0(x1)))) -> 5^1(2(1(0(4(x1))))) 37.81/10.57 5^1(4(1(0(x1)))) -> 1^1(0(4(x1))) 37.81/10.57 1^1(5(1(0(x1)))) -> 1^1(0(5(2(2(1(x1)))))) 37.81/10.57 1^1(5(1(0(x1)))) -> 5^1(2(2(1(x1)))) 37.81/10.57 1^1(5(1(0(x1)))) -> 1^1(x1) 37.81/10.57 3^1(5(1(0(x1)))) -> 1^1(2(3(5(0(x1))))) 37.81/10.57 3^1(5(1(0(x1)))) -> 3^1(5(0(x1))) 37.81/10.57 3^1(5(1(0(x1)))) -> 5^1(0(x1)) 37.81/10.57 1^1(4(2(0(x1)))) -> 3^1(3(1(x1))) 37.81/10.57 1^1(4(2(0(x1)))) -> 3^1(1(x1)) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(2(4(x1))) 37.81/10.57 5^1(4(2(0(x1)))) -> 5^1(2(2(0(x1)))) 37.81/10.57 5^1(1(3(0(x1)))) -> 3^1(5(2(1(0(x1))))) 37.81/10.57 5^1(1(3(0(x1)))) -> 5^1(2(1(0(x1)))) 37.81/10.57 5^1(1(3(0(x1)))) -> 1^1(0(x1)) 37.81/10.57 5^1(1(3(0(x1)))) -> 3^1(5(2(1(x1)))) 37.81/10.57 5^1(1(3(0(x1)))) -> 5^1(2(1(x1))) 37.81/10.57 5^1(1(3(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(3(0(x1)))) -> 3^1(5(2(1(x1)))) 37.81/10.57 1^1(5(3(0(x1)))) -> 5^1(2(1(x1))) 37.81/10.57 1^1(5(3(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(3(0(x1)))) -> 3^1(2(1(2(5(0(x1)))))) 37.81/10.57 1^1(5(3(0(x1)))) -> 1^1(2(5(0(x1)))) 37.81/10.57 1^1(5(3(0(x1)))) -> 5^1(0(x1)) 37.81/10.57 5^1(5(3(0(x1)))) -> 5^1(5(2(3(0(x1))))) 37.81/10.57 5^1(5(3(0(x1)))) -> 5^1(2(3(0(x1)))) 37.81/10.57 1^1(0(4(0(x1)))) -> 1^1(0(4(4(0(2(x1)))))) 37.81/10.57 5^1(1(4(0(x1)))) -> 5^1(2(1(x1))) 37.81/10.57 5^1(1(4(0(x1)))) -> 1^1(x1) 37.81/10.57 5^1(3(4(0(x1)))) -> 5^1(2(3(4(0(x1))))) 37.81/10.57 1^1(5(4(0(x1)))) -> 1^1(0(4(4(5(2(x1)))))) 37.81/10.57 1^1(5(4(0(x1)))) -> 5^1(2(x1)) 37.81/10.57 5^1(1(0(3(x1)))) -> 5^1(0(4(1(3(x1))))) 37.81/10.57 5^1(1(0(3(x1)))) -> 1^1(3(x1)) 37.81/10.57 1^1(3(0(3(x1)))) -> 3^1(3(1(x1))) 37.81/10.57 1^1(3(0(3(x1)))) -> 3^1(1(x1)) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 5^1(3(0(3(x1)))) -> 3^1(2(0(5(2(3(x1)))))) 37.81/10.57 5^1(3(0(3(x1)))) -> 5^1(2(3(x1))) 37.81/10.57 1^1(0(3(3(x1)))) -> 3^1(2(2(3(1(0(x1)))))) 37.81/10.57 1^1(0(3(3(x1)))) -> 3^1(1(0(x1))) 37.81/10.57 1^1(0(3(3(x1)))) -> 1^1(0(x1)) 37.81/10.57 1^1(5(4(3(x1)))) -> 1^1(2(4(5(2(3(x1)))))) 37.81/10.57 1^1(5(4(3(x1)))) -> 5^1(2(3(x1))) 37.81/10.57 5^1(3(1(4(x1)))) -> 3^1(5(2(1(x1)))) 37.81/10.57 5^1(3(1(4(x1)))) -> 5^1(2(1(x1))) 37.81/10.57 5^1(3(1(4(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(1(4(x1)))) -> 1^1(2(1(5(4(4(x1)))))) 37.81/10.57 1^1(5(1(4(x1)))) -> 1^1(5(4(4(x1)))) 37.81/10.57 1^1(5(1(4(x1)))) -> 5^1(4(4(x1))) 37.81/10.57 5^1(1(4(4(x1)))) -> 5^1(2(1(4(x1)))) 37.81/10.57 5^1(1(4(4(x1)))) -> 1^1(4(x1)) 37.81/10.57 5^1(5(4(1(0(x1))))) -> 5^1(2(4(1(5(0(x1)))))) 37.81/10.57 5^1(5(4(1(0(x1))))) -> 1^1(5(0(x1))) 37.81/10.57 5^1(5(4(1(0(x1))))) -> 5^1(0(x1)) 37.81/10.57 5^1(4(1(2(0(x1))))) -> 5^1(2(1(0(0(x1))))) 37.81/10.57 5^1(4(1(2(0(x1))))) -> 1^1(0(0(x1))) 37.81/10.57 5^1(5(1(2(0(x1))))) -> 5^1(5(2(2(1(0(x1)))))) 37.81/10.57 5^1(5(1(2(0(x1))))) -> 5^1(2(2(1(0(x1))))) 37.81/10.57 5^1(5(1(2(0(x1))))) -> 1^1(0(x1)) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 3^1(1(x1)) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 3^1(4(5(4(0(x1))))) -> 3^1(4(4(0(5(2(x1)))))) 37.81/10.57 3^1(4(5(4(0(x1))))) -> 5^1(2(x1)) 37.81/10.57 1^1(5(1(5(0(x1))))) -> 5^1(2(1(1(5(0(x1)))))) 37.81/10.57 1^1(5(1(5(0(x1))))) -> 1^1(1(5(0(x1)))) 37.81/10.57 5^1(1(2(5(0(x1))))) -> 5^1(5(2(1(x1)))) 37.81/10.57 5^1(1(2(5(0(x1))))) -> 5^1(2(1(x1))) 37.81/10.57 5^1(1(2(5(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(2(5(0(x1))))) -> 1^1(2(5(4(x1)))) 37.81/10.57 1^1(4(2(5(0(x1))))) -> 5^1(4(x1)) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(3(1(4(4(0(x1)))))) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 3^1(1(4(4(0(x1))))) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(4(4(0(x1)))) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(0(2(3(4(x1))))) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 3^1(4(x1)) 37.81/10.57 5^1(5(3(0(3(x1))))) -> 5^1(0(5(2(3(3(x1)))))) 37.81/10.57 5^1(5(3(0(3(x1))))) -> 5^1(2(3(3(x1)))) 37.81/10.57 5^1(5(3(0(3(x1))))) -> 3^1(3(x1)) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 5^1(2(3(3(0(1(x1)))))) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 3^1(3(0(1(x1)))) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 3^1(0(1(x1))) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(4(x1))))) -> 1^1(3(1(0(4(4(x1)))))) 37.81/10.57 1^1(4(1(0(4(x1))))) -> 3^1(1(0(4(4(x1))))) 37.81/10.57 1^1(4(1(0(4(x1))))) -> 1^1(0(4(4(x1)))) 37.81/10.57 1^1(5(1(0(4(x1))))) -> 1^1(4(5(2(1(0(x1)))))) 37.81/10.57 1^1(5(1(0(4(x1))))) -> 5^1(2(1(0(x1)))) 37.81/10.57 1^1(5(1(0(4(x1))))) -> 1^1(0(x1)) 37.81/10.57 5^1(4(2(0(4(x1))))) -> 5^1(2(0(4(x1)))) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(4(5(2(1(1(x1)))))) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 5^1(2(1(1(x1)))) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(1(x1)) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(5(4(x1))))) -> 5^1(1(2(1(4(4(x1)))))) 37.81/10.57 1^1(4(1(5(4(x1))))) -> 1^1(2(1(4(4(x1))))) 37.81/10.57 1^1(4(1(5(4(x1))))) -> 1^1(4(4(x1))) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 5^1(2(2(1(3(4(x1)))))) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 1^1(3(4(x1))) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 3^1(4(x1)) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 3^1(5(2(1(4(x1))))) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 5^1(2(1(4(x1)))) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 1^1(4(x1)) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 5^1(5(2(3(1(x1))))) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 5^1(2(3(1(x1)))) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 3^1(1(x1)) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 1^1(x1) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(1(3(0(x1)))) -> 3(5(2(1(0(x1))))) 37.81/10.57 5(1(3(0(x1)))) -> 4(0(3(5(2(1(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 5(5(3(0(x1)))) -> 5(5(2(3(0(x1))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 5(1(4(0(x1)))) -> 4(0(5(2(1(x1))))) 37.81/10.57 5(3(4(0(x1)))) -> 4(5(2(3(4(0(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 5(1(0(3(x1)))) -> 4(5(0(4(1(3(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 5(3(0(3(x1)))) -> 3(2(0(5(2(3(x1)))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 5(3(1(4(x1)))) -> 4(4(3(5(2(1(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 5(1(4(4(x1)))) -> 4(5(2(1(4(x1))))) 37.81/10.57 5(5(4(1(0(x1))))) -> 5(2(4(1(5(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(5(1(2(0(x1))))) -> 5(5(2(2(1(0(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 5(1(2(5(0(x1))))) -> 4(0(5(5(2(1(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 5(5(3(0(3(x1))))) -> 5(0(5(2(3(3(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (5) DependencyGraphProof (EQUIVALENT) 37.81/10.57 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 105 less nodes. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (6) 37.81/10.57 Complex Obligation (AND) 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (7) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(0(3(3(x1)))) -> 1^1(0(x1)) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(1(3(0(x1)))) -> 3(5(2(1(0(x1))))) 37.81/10.57 5(1(3(0(x1)))) -> 4(0(3(5(2(1(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 5(5(3(0(x1)))) -> 5(5(2(3(0(x1))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 5(1(4(0(x1)))) -> 4(0(5(2(1(x1))))) 37.81/10.57 5(3(4(0(x1)))) -> 4(5(2(3(4(0(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 5(1(0(3(x1)))) -> 4(5(0(4(1(3(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 5(3(0(3(x1)))) -> 3(2(0(5(2(3(x1)))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 5(3(1(4(x1)))) -> 4(4(3(5(2(1(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 5(1(4(4(x1)))) -> 4(5(2(1(4(x1))))) 37.81/10.57 5(5(4(1(0(x1))))) -> 5(2(4(1(5(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(5(1(2(0(x1))))) -> 5(5(2(2(1(0(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 5(1(2(5(0(x1))))) -> 4(0(5(5(2(1(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 5(5(3(0(3(x1))))) -> 5(0(5(2(3(3(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (8) UsableRulesProof (EQUIVALENT) 37.81/10.57 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. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (9) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(0(3(3(x1)))) -> 1^1(0(x1)) 37.81/10.57 37.81/10.57 R is empty. 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (10) QDPOrderProof (EQUIVALENT) 37.81/10.57 We use the reduction pair processor [LPAR04,JAR06]. 37.81/10.57 37.81/10.57 37.81/10.57 The following pairs can be oriented strictly and are deleted. 37.81/10.57 37.81/10.57 1^1(0(3(3(x1)))) -> 1^1(0(x1)) 37.81/10.57 The remaining pairs can at least be oriented weakly. 37.81/10.57 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 37.81/10.57 37.81/10.57 POL( 1^1_1(x_1) ) = max{0, 2x_1 - 2} 37.81/10.57 POL( 0_1(x_1) ) = x_1 37.81/10.57 POL( 3_1(x_1) ) = 2x_1 + 1 37.81/10.57 37.81/10.57 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 37.81/10.57 none 37.81/10.57 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (11) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 P is empty. 37.81/10.57 R is empty. 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (12) PisEmptyProof (EQUIVALENT) 37.81/10.57 The TRS P is empty. Hence, there is no (P,Q,R) chain. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (13) 37.81/10.57 YES 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (14) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(5(1(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(3(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 1^1(x1) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(1(x1)) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(x1) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 1^1(3(4(x1))) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 1^1(4(x1)) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 1^1(x1) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(1(3(0(x1)))) -> 3(5(2(1(0(x1))))) 37.81/10.57 5(1(3(0(x1)))) -> 4(0(3(5(2(1(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 5(5(3(0(x1)))) -> 5(5(2(3(0(x1))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 5(1(4(0(x1)))) -> 4(0(5(2(1(x1))))) 37.81/10.57 5(3(4(0(x1)))) -> 4(5(2(3(4(0(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 5(1(0(3(x1)))) -> 4(5(0(4(1(3(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 5(3(0(3(x1)))) -> 3(2(0(5(2(3(x1)))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 5(3(1(4(x1)))) -> 4(4(3(5(2(1(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 5(1(4(4(x1)))) -> 4(5(2(1(4(x1))))) 37.81/10.57 5(5(4(1(0(x1))))) -> 5(2(4(1(5(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(5(1(2(0(x1))))) -> 5(5(2(2(1(0(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 5(1(2(5(0(x1))))) -> 4(0(5(5(2(1(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 5(5(3(0(3(x1))))) -> 5(0(5(2(3(3(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (15) UsableRulesProof (EQUIVALENT) 37.81/10.57 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. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (16) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(5(1(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(3(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 1^1(x1) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(1(x1)) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(x1) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 1^1(3(4(x1))) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 1^1(4(x1)) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 1^1(x1) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (17) QDPOrderProof (EQUIVALENT) 37.81/10.57 We use the reduction pair processor [LPAR04,JAR06]. 37.81/10.57 37.81/10.57 37.81/10.57 The following pairs can be oriented strictly and are deleted. 37.81/10.57 37.81/10.57 1^1(5(1(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(5(3(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(5(0(3(x1))))) -> 1^1(x1) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(1(x1)) 37.81/10.57 1^1(5(1(1(4(x1))))) -> 1^1(x1) 37.81/10.57 1^1(3(2(5(4(x1))))) -> 1^1(3(4(x1))) 37.81/10.57 1^1(3(4(5(4(x1))))) -> 1^1(4(x1)) 37.81/10.57 1^1(3(5(5(4(x1))))) -> 1^1(x1) 37.81/10.57 The remaining pairs can at least be oriented weakly. 37.81/10.57 Used ordering: Polynomial interpretation [POLO]: 37.81/10.57 37.81/10.57 POL(0(x_1)) = x_1 37.81/10.57 POL(1(x_1)) = x_1 37.81/10.57 POL(1^1(x_1)) = x_1 37.81/10.57 POL(2(x_1)) = x_1 37.81/10.57 POL(3(x_1)) = x_1 37.81/10.57 POL(4(x_1)) = x_1 37.81/10.57 POL(5(x_1)) = 1 + x_1 37.81/10.57 37.81/10.57 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 37.81/10.57 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (18) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 1(1(0(x1))) -> 0(2(1(2(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(2(3(1(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(1(2(3(x1))))) 37.81/10.57 1(3(0(x1))) -> 0(2(3(3(3(1(x1)))))) 37.81/10.57 1(4(0(x1))) -> 4(0(2(1(2(x1))))) 37.81/10.57 1(4(1(0(x1)))) -> 1(4(2(2(1(0(x1)))))) 37.81/10.57 1(5(1(0(x1)))) -> 1(0(5(2(2(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(3(3(1(x1)))))) 37.81/10.57 1(4(2(0(x1)))) -> 4(0(2(1(2(4(x1)))))) 37.81/10.57 1(5(3(0(x1)))) -> 0(3(5(2(1(x1))))) 37.81/10.57 1(5(3(0(x1)))) -> 3(2(1(2(5(0(x1)))))) 37.81/10.57 1(0(4(0(x1)))) -> 1(0(4(4(0(2(x1)))))) 37.81/10.57 1(5(4(0(x1)))) -> 1(0(4(4(5(2(x1)))))) 37.81/10.57 1(3(0(3(x1)))) -> 0(2(3(3(1(x1))))) 37.81/10.57 1(0(3(3(x1)))) -> 3(2(2(3(1(0(x1)))))) 37.81/10.57 1(5(4(3(x1)))) -> 1(2(4(5(2(3(x1)))))) 37.81/10.57 1(5(1(4(x1)))) -> 1(2(1(5(4(4(x1)))))) 37.81/10.57 1(4(2(4(0(x1))))) -> 4(4(0(2(3(1(x1)))))) 37.81/10.57 1(5(1(5(0(x1))))) -> 5(2(1(1(5(0(x1)))))) 37.81/10.57 1(4(2(5(0(x1))))) -> 0(2(1(2(5(4(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(3(1(4(4(0(x1)))))) 37.81/10.57 1(4(1(0(3(x1))))) -> 1(1(0(2(3(4(x1)))))) 37.81/10.57 1(3(5(0(3(x1))))) -> 5(2(3(3(0(1(x1)))))) 37.81/10.57 1(4(1(0(4(x1))))) -> 1(3(1(0(4(4(x1)))))) 37.81/10.57 1(5(1(0(4(x1))))) -> 1(4(5(2(1(0(x1)))))) 37.81/10.57 1(5(1(1(4(x1))))) -> 1(4(5(2(1(1(x1)))))) 37.81/10.57 1(4(1(5(4(x1))))) -> 5(1(2(1(4(4(x1)))))) 37.81/10.57 1(3(2(5(4(x1))))) -> 5(2(2(1(3(4(x1)))))) 37.81/10.57 1(3(4(5(4(x1))))) -> 4(3(5(2(1(4(x1)))))) 37.81/10.57 1(3(5(5(4(x1))))) -> 4(5(5(2(3(1(x1)))))) 37.81/10.57 3(5(1(0(x1)))) -> 1(2(3(5(0(x1))))) 37.81/10.57 5(4(0(0(x1)))) -> 4(5(2(0(0(x1))))) 37.81/10.57 5(4(1(0(x1)))) -> 4(5(2(1(0(4(x1)))))) 37.81/10.57 5(4(2(0(x1)))) -> 4(0(5(2(2(0(x1)))))) 37.81/10.57 5(4(1(2(0(x1))))) -> 4(5(2(1(0(0(x1)))))) 37.81/10.57 5(4(2(0(4(x1))))) -> 4(0(5(2(0(4(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (19) UsableRulesProof (EQUIVALENT) 37.81/10.57 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. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (20) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 The TRS P consists of the following rules: 37.81/10.57 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 37.81/10.57 The TRS R consists of the following rules: 37.81/10.57 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (21) MRRProof (EQUIVALENT) 37.81/10.57 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 37.81/10.57 37.81/10.57 Strictly oriented dependency pairs: 37.81/10.57 37.81/10.57 1^1(3(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(1(0(x1))) -> 1^1(x1) 37.81/10.57 1^1(4(2(0(x1)))) -> 1^1(x1) 37.81/10.57 1^1(3(0(3(x1)))) -> 1^1(x1) 37.81/10.57 1^1(4(2(4(0(x1))))) -> 1^1(x1) 37.81/10.57 1^1(4(1(0(3(x1))))) -> 1^1(1(0(2(3(4(x1)))))) 37.81/10.57 37.81/10.57 Strictly oriented rules of the TRS R: 37.81/10.57 37.81/10.57 3(4(5(4(0(x1))))) -> 3(4(4(0(5(2(x1)))))) 37.81/10.57 37.81/10.57 Used ordering: Polynomial interpretation [POLO]: 37.81/10.57 37.81/10.57 POL(0(x_1)) = 3 + x_1 37.81/10.57 POL(1(x_1)) = 2*x_1 37.81/10.57 POL(1^1(x_1)) = 3*x_1 37.81/10.57 POL(2(x_1)) = 1 + x_1 37.81/10.57 POL(3(x_1)) = 3 + 2*x_1 37.81/10.57 POL(4(x_1)) = 1 + 2*x_1 37.81/10.57 POL(5(x_1)) = 2 + 2*x_1 37.81/10.57 37.81/10.57 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (22) 37.81/10.57 Obligation: 37.81/10.57 Q DP problem: 37.81/10.57 P is empty. 37.81/10.57 R is empty. 37.81/10.57 Q is empty. 37.81/10.57 We have to consider all minimal (P,Q,R)-chains. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (23) PisEmptyProof (EQUIVALENT) 37.81/10.57 The TRS P is empty. Hence, there is no (P,Q,R) chain. 37.81/10.57 ---------------------------------------- 37.81/10.57 37.81/10.57 (24) 37.81/10.57 YES 38.09/10.69 EOF