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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 24 ms] (2) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(d(x1)) -> d(b(c(b(d(x1))))) a(x1) -> b(b(f(b(b(x1))))) b(d(b(x1))) -> a(d(x1)) d(f(x1)) -> b(d(x1)) Q is empty. ---------------------------------------- (1) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 4. This implies Q-termination of R. The following rules were used to construct the certificate: a(d(x1)) -> d(b(c(b(d(x1))))) a(x1) -> b(b(f(b(b(x1))))) b(d(b(x1))) -> a(d(x1)) d(f(x1)) -> b(d(x1)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 136, 137, 138, 139, 140, 141, 142, 143 Node 1 is start node and node 2 is final node. Those nodes are connected through the following edges: * 1 to 3 labelled d_1(0)* 1 to 7 labelled b_1(0)* 1 to 6 labelled a_1(0), b_1(0)* 1 to 11 labelled b_1(1)* 1 to 15 labelled d_1(1)* 1 to 18 labelled a_1(1)* 1 to 19 labelled b_1(2)* 1 to 23 labelled d_1(2)* 1 to 86 labelled a_1(1)* 1 to 87 labelled b_1(2)* 1 to 91 labelled d_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled b_1(0)* 4 to 5 labelled c_1(0)* 5 to 6 labelled b_1(0)* 5 to 18 labelled a_1(1)* 5 to 19 labelled b_1(2)* 5 to 23 labelled d_1(2)* 5 to 86 labelled a_1(1)* 5 to 87 labelled b_1(2)* 5 to 91 labelled d_1(2)* 6 to 2 labelled d_1(0)* 6 to 18 labelled b_1(1), a_1(1)* 6 to 19 labelled b_1(2)* 6 to 23 labelled d_1(2)* 6 to 94 labelled a_1(2)* 6 to 95 labelled b_1(3)* 6 to 99 labelled d_1(3)* 6 to 91 labelled d_1(2)* 7 to 8 labelled b_1(0)* 8 to 9 labelled f_1(0)* 9 to 10 labelled b_1(0)* 9 to 86 labelled a_1(1)* 9 to 87 labelled b_1(2)* 9 to 91 labelled d_1(2)* 9 to 18 labelled a_1(1)* 9 to 19 labelled b_1(2)* 9 to 23 labelled d_1(2)* 10 to 2 labelled b_1(0)* 10 to 18 labelled a_1(1)* 10 to 19 labelled b_1(2)* 10 to 23 labelled d_1(2)* 10 to 91 labelled d_1(2)* 11 to 12 labelled b_1(1)* 12 to 13 labelled f_1(1)* 13 to 14 labelled b_1(1)* 13 to 94 labelled a_1(2)* 13 to 95 labelled b_1(3)* 13 to 99 labelled d_1(3)* 14 to 6 labelled b_1(1)* 14 to 18 labelled a_1(1)* 14 to 19 labelled b_1(2)* 14 to 23 labelled d_1(2)* 14 to 94 labelled a_1(2)* 14 to 95 labelled b_1(3)* 14 to 99 labelled d_1(3)* 14 to 91 labelled d_1(2)* 15 to 16 labelled b_1(1)* 16 to 17 labelled c_1(1)* 17 to 18 labelled b_1(1), a_1(1)* 17 to 19 labelled b_1(2)* 17 to 23 labelled d_1(2)* 17 to 94 labelled a_1(2)* 17 to 95 labelled b_1(3)* 17 to 99 labelled d_1(3)* 17 to 91 labelled d_1(2)* 18 to 2 labelled d_1(1)* 18 to 18 labelled b_1(1), a_1(1)* 18 to 19 labelled b_1(2)* 18 to 23 labelled d_1(1), d_1(2)* 18 to 94 labelled a_1(2)* 18 to 95 labelled b_1(3)* 18 to 99 labelled d_1(3), d_1(1)* 18 to 100 labelled d_1(1)* 18 to 91 labelled d_1(2), d_1(1)* 18 to 92 labelled d_1(1)* 19 to 20 labelled b_1(2)* 20 to 21 labelled f_1(2)* 21 to 22 labelled b_1(2)* 21 to 102 labelled a_1(3)* 21 to 136 labelled b_1(4)* 21 to 140 labelled d_1(4)* 22 to 18 labelled b_1(2), a_1(1)* 22 to 19 labelled b_1(2)* 22 to 23 labelled d_1(2)* 22 to 94 labelled a_1(2)* 22 to 102 labelled a_1(3)* 22 to 95 labelled b_1(3)* 22 to 136 labelled b_1(4)* 22 to 99 labelled d_1(3)* 22 to 140 labelled d_1(4)* 22 to 91 labelled d_1(2)* 23 to 24 labelled b_1(2)* 24 to 25 labelled c_1(2)* 25 to 26 labelled b_1(2)* 25 to 18 labelled a_1(1)* 25 to 19 labelled b_1(2)* 25 to 23 labelled d_1(2)* 25 to 102 labelled a_1(3)* 25 to 136 labelled b_1(4)* 25 to 140 labelled d_1(4)* 25 to 91 labelled d_1(2)* 26 to 2 labelled d_1(2)* 26 to 18 labelled b_1(1), a_1(1)* 26 to 19 labelled b_1(2)* 26 to 23 labelled d_1(2)* 26 to 94 labelled a_1(2)* 26 to 95 labelled b_1(3)* 26 to 99 labelled d_1(3), d_1(2)* 26 to 91 labelled d_1(2)* 86 to 24 labelled d_1(1)* 87 to 88 labelled b_1(2)* 88 to 89 labelled f_1(2)* 89 to 90 labelled b_1(2)* 90 to 86 labelled b_1(2)* 91 to 92 labelled b_1(2)* 92 to 93 labelled c_1(2)* 93 to 94 labelled b_1(2)* 94 to 24 labelled d_1(2)* 94 to 100 labelled d_1(2)* 94 to 92 labelled d_1(2)* 95 to 96 labelled b_1(3)* 96 to 97 labelled f_1(3)* 97 to 98 labelled b_1(3)* 98 to 94 labelled b_1(3)* 99 to 100 labelled b_1(3)* 100 to 101 labelled c_1(3)* 101 to 102 labelled b_1(3)* 102 to 24 labelled d_1(3)* 102 to 100 labelled d_1(3)* 102 to 141 labelled d_1(3)* 102 to 92 labelled d_1(3)* 136 to 137 labelled b_1(4)* 137 to 138 labelled f_1(4)* 138 to 139 labelled b_1(4)* 139 to 102 labelled b_1(4)* 140 to 141 labelled b_1(4)* 141 to 142 labelled c_1(4)* 142 to 143 labelled b_1(4)* 143 to 24 labelled d_1(4)* 143 to 100 labelled d_1(4)* 143 to 141 labelled d_1(4)* 143 to 92 labelled d_1(4) ---------------------------------------- (2) YES