YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: half(0) -> 0 half(s(s(x))) -> s(half(x)) log(s(0)) -> 0 log(s(s(x))) -> s(log(s(half(x)))) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0'(half(x)) -> 0'(x) s(s(half(x))) -> half(s(x)) 0'(s(log(x))) -> 0'(x) s(s(log(x))) -> half(s(log(s(x)))) Q is empty. ---------------------------------------- (3) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R. The following rules were used to construct the certificate: 0'(half(x)) -> 0'(x) s(s(half(x))) -> half(s(x)) 0'(s(log(x))) -> 0'(x) s(s(log(x))) -> half(s(log(s(x)))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 Node 35 is start node and node 36 is final node. Those nodes are connected through the following edges: * 35 to 36 labelled 0'_1(0), 0'_1(1)* 35 to 37 labelled half_1(0)* 35 to 38 labelled half_1(0)* 36 to 36 labelled #_1(0)* 37 to 36 labelled s_1(0)* 37 to 41 labelled half_1(1)* 37 to 42 labelled half_1(1)* 38 to 39 labelled s_1(0)* 39 to 40 labelled log_1(0)* 40 to 36 labelled s_1(0)* 40 to 41 labelled half_1(1)* 40 to 42 labelled half_1(1)* 41 to 36 labelled s_1(1)* 41 to 41 labelled half_1(1)* 41 to 42 labelled half_1(1)* 42 to 43 labelled s_1(1)* 43 to 44 labelled log_1(1)* 44 to 36 labelled s_1(1)* 44 to 41 labelled half_1(1)* 44 to 42 labelled half_1(1) ---------------------------------------- (4) YES