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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 10 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: t(o(x1)) -> m(a(x1)) t(e(x1)) -> n(s(x1)) a(l(x1)) -> a(t(x1)) o(m(a(x1))) -> t(e(n(x1))) s(a(x1)) -> l(a(t(o(m(a(t(e(x1)))))))) n(s(x1)) -> a(l(a(t(x1)))) 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: o(t(x1)) -> a(m(x1)) e(t(x1)) -> s(n(x1)) l(a(x1)) -> t(a(x1)) a(m(o(x1))) -> n(e(t(x1))) a(s(x1)) -> e(t(a(m(o(t(a(l(x1)))))))) s(n(x1)) -> t(a(l(a(x1)))) Q is empty. ---------------------------------------- (3) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R. The following rules were used to construct the certificate: o(t(x1)) -> a(m(x1)) e(t(x1)) -> s(n(x1)) l(a(x1)) -> t(a(x1)) a(m(o(x1))) -> n(e(t(x1))) a(s(x1)) -> e(t(a(m(o(t(a(l(x1)))))))) s(n(x1)) -> t(a(l(a(x1)))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 53, 54, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 132, 133, 134, 135, 136, 137, 143 Node 53 is start node and node 54 is final node. Those nodes are connected through the following edges: * 53 to 68 labelled a_1(0)* 53 to 69 labelled s_1(0)* 53 to 70 labelled t_1(0)* 53 to 71 labelled n_1(0)* 53 to 73 labelled e_1(0)* 53 to 80 labelled t_1(0)* 53 to 83 labelled t_1(1)* 53 to 86 labelled s_1(1)* 53 to 87 labelled n_1(1)* 53 to 100 labelled t_1(2)* 54 to 54 labelled #_1(0)* 68 to 54 labelled m_1(0)* 69 to 54 labelled n_1(0)* 70 to 54 labelled a_1(0)* 70 to 87 labelled n_1(1)* 70 to 89 labelled e_1(1)* 70 to 103 labelled s_1(2)* 70 to 112 labelled t_1(3)* 71 to 72 labelled e_1(0)* 71 to 96 labelled s_1(1)* 71 to 104 labelled t_1(2)* 72 to 54 labelled t_1(0)* 73 to 74 labelled t_1(0)* 74 to 75 labelled a_1(0)* 74 to 98 labelled n_1(1)* 75 to 76 labelled m_1(0)* 76 to 77 labelled o_1(0)* 76 to 97 labelled a_1(1)* 77 to 78 labelled t_1(0)* 78 to 79 labelled a_1(0)* 79 to 54 labelled l_1(0)* 79 to 85 labelled t_1(1)* 80 to 81 labelled a_1(0)* 81 to 82 labelled l_1(0)* 81 to 85 labelled t_1(1)* 82 to 54 labelled a_1(0)* 82 to 87 labelled n_1(1)* 82 to 89 labelled e_1(1)* 82 to 103 labelled s_1(2)* 82 to 112 labelled t_1(3)* 83 to 84 labelled a_1(1)* 84 to 85 labelled l_1(1)* 84 to 106 labelled t_1(2)* 85 to 54 labelled a_1(1)* 85 to 87 labelled n_1(1)* 85 to 89 labelled e_1(1)* 85 to 103 labelled s_1(2)* 85 to 112 labelled t_1(3)* 86 to 74 labelled n_1(1)* 87 to 88 labelled e_1(1)* 87 to 107 labelled s_1(2)* 87 to 115 labelled t_1(3)* 88 to 54 labelled t_1(1)* 89 to 90 labelled t_1(1)* 90 to 91 labelled a_1(1)* 90 to 110 labelled n_1(2)* 91 to 92 labelled m_1(1)* 92 to 93 labelled o_1(1)* 92 to 108 labelled a_1(2)* 93 to 94 labelled t_1(1)* 94 to 95 labelled a_1(1)* 95 to 54 labelled l_1(1)* 95 to 85 labelled t_1(1)* 96 to 54 labelled n_1(1)* 97 to 78 labelled m_1(1)* 98 to 99 labelled e_1(1)* 98 to 109 labelled s_1(2)* 98 to 118 labelled t_1(3)* 99 to 77 labelled t_1(1)* 100 to 101 labelled a_1(2)* 101 to 102 labelled l_1(2)* 101 to 121 labelled t_1(3)* 102 to 74 labelled a_1(2)* 103 to 90 labelled n_1(2)* 104 to 105 labelled a_1(2)* 105 to 106 labelled l_1(2)* 105 to 117 labelled t_1(3)* 106 to 54 labelled a_1(2)* 106 to 87 labelled n_1(1)* 106 to 89 labelled e_1(1)* 106 to 103 labelled s_1(2)* 106 to 112 labelled t_1(3)* 107 to 54 labelled n_1(2)* 108 to 94 labelled m_1(2)* 109 to 77 labelled n_1(2)* 110 to 111 labelled e_1(2)* 110 to 122 labelled s_1(3)* 110 to 132 labelled t_1(4)* 111 to 93 labelled t_1(2)* 112 to 113 labelled a_1(3)* 113 to 114 labelled l_1(3)* 113 to 135 labelled t_1(4)* 114 to 90 labelled a_1(3)* 115 to 116 labelled a_1(3)* 116 to 117 labelled l_1(3)* 116 to 136 labelled t_1(4)* 117 to 54 labelled a_1(3)* 117 to 87 labelled n_1(1)* 117 to 89 labelled e_1(1)* 117 to 103 labelled s_1(2)* 117 to 112 labelled t_1(3)* 118 to 119 labelled a_1(3)* 119 to 120 labelled l_1(3)* 119 to 137 labelled t_1(4)* 120 to 77 labelled a_1(3)* 121 to 74 labelled a_1(3)* 122 to 93 labelled n_1(3)* 132 to 133 labelled a_1(4)* 133 to 134 labelled l_1(4)* 133 to 143 labelled t_1(5)* 134 to 93 labelled a_1(4)* 135 to 90 labelled a_1(4)* 136 to 54 labelled a_1(4)* 136 to 87 labelled n_1(1)* 136 to 89 labelled e_1(1)* 136 to 103 labelled s_1(2)* 136 to 112 labelled t_1(3)* 137 to 77 labelled a_1(4)* 143 to 93 labelled a_1(5) ---------------------------------------- (4) YES