Termination w.r.t. Q of the following Term Rewriting System could be proven:
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.
↳ QTRS
↳ RFCMatchBoundsTRSProof
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.
Termination of the TRS R could be shown with a Match Bound [6,7] of 6. This implies Q-termination of R.
The following rules were used to construct the certificate:
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))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 9, 5, 6, 8, 4, 7, 3, 10, 11, 12, 13, 14, 17, 16, 15, 18, 19, 20, 23, 21, 22, 24, 31, 27, 28, 30, 26, 29, 25, 32, 33, 34, 35, 36, 37, 40, 39, 38, 43, 42, 41, 46, 47, 50, 49, 45, 48, 44, 51, 52, 53, 54, 55, 56, 57, 60, 58, 59, 63, 61, 62, 64, 65, 66, 67, 68, 71, 70, 69, 74, 73, 72, 86, 87, 90, 88, 89, 91
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled l_1(0)
- 1 to 10 labelled m_1(0)
- 1 to 11 labelled t_1(0)
- 1 to 13 labelled a_1(0)
- 1 to 14 labelled n_1(0)
- 1 to 15 labelled a_1(0)
- 1 to 19 labelled n_1(1)
- 1 to 21 labelled a_1(1)
- 1 to 34 labelled a_1(2)
- 1 to 41 labelled a_1(2)
- 1 to 53 labelled a_1(3)
- 2 to 2 labelled #_1(0)
- 9 to 2 labelled e_1(0)
- 5 to 6 labelled o_1(0)
- 5 to 32 labelled t_1(1)
- 5 to 36 labelled n_1(2)
- 5 to 61 labelled a_1(3)
- 5 to 67 labelled a_1(4)
- 6 to 7 labelled m_1(0)
- 8 to 9 labelled t_1(0)
- 8 to 18 labelled n_1(1)
- 8 to 38 labelled a_1(2)
- 8 to 54 labelled a_1(3)
- 4 to 5 labelled t_1(0)
- 4 to 20 labelled m_1(1)
- 7 to 8 labelled a_1(0)
- 3 to 4 labelled a_1(0)
- 10 to 2 labelled a_1(0)
- 10 to 23 labelled a_1(1)
- 11 to 12 labelled e_1(0)
- 12 to 2 labelled n_1(0)
- 12 to 21 labelled a_1(1)
- 12 to 34 labelled a_1(2)
- 13 to 2 labelled t_1(0)
- 13 to 24 labelled m_1(1)
- 13 to 18 labelled n_1(1)
- 13 to 38 labelled a_1(2)
- 13 to 54 labelled a_1(3)
- 14 to 2 labelled s_1(0)
- 14 to 25 labelled l_1(1)
- 17 to 2 labelled t_1(0)
- 17 to 24 labelled m_1(1)
- 17 to 18 labelled n_1(1)
- 17 to 38 labelled a_1(2)
- 17 to 54 labelled a_1(3)
- 16 to 17 labelled a_1(0)
- 15 to 16 labelled l_1(0)
- 18 to 2 labelled s_1(1)
- 18 to 25 labelled l_1(1)
- 19 to 12 labelled s_1(1)
- 19 to 44 labelled l_1(2)
- 20 to 6 labelled a_1(1)
- 23 to 2 labelled t_1(1)
- 23 to 24 labelled m_1(1)
- 23 to 18 labelled n_1(1)
- 23 to 38 labelled a_1(2)
- 23 to 54 labelled a_1(3)
- 21 to 22 labelled l_1(1)
- 21 to 16 labelled t_1(1)
- 22 to 23 labelled a_1(1)
- 24 to 2 labelled a_1(1)
- 24 to 23 labelled a_1(1)
- 31 to 2 labelled e_1(1)
- 27 to 28 labelled o_1(1)
- 27 to 51 labelled t_1(2)
- 27 to 57 labelled n_1(3)
- 27 to 72 labelled a_1(4)
- 27 to 87 labelled a_1(5)
- 28 to 29 labelled m_1(1)
- 30 to 31 labelled t_1(1)
- 30 to 35 labelled n_1(2)
- 30 to 58 labelled a_1(3)
- 30 to 66 labelled a_1(4)
- 26 to 27 labelled t_1(1)
- 26 to 37 labelled m_1(2)
- 29 to 30 labelled a_1(1)
- 25 to 26 labelled a_1(1)
- 32 to 33 labelled e_1(1)
- 33 to 8 labelled n_1(1)
- 34 to 22 labelled t_1(2)
- 35 to 2 labelled s_1(2)
- 35 to 25 labelled l_1(1)
- 36 to 33 labelled s_1(2)
- 37 to 28 labelled a_1(2)
- 40 to 2 labelled t_1(2)
- 40 to 24 labelled m_1(1)
- 40 to 18 labelled n_1(1)
- 40 to 38 labelled a_1(2)
- 40 to 54 labelled a_1(3)
- 39 to 40 labelled a_1(2)
- 38 to 39 labelled l_1(2)
- 43 to 12 labelled t_1(2)
- 42 to 43 labelled a_1(2)
- 41 to 42 labelled l_1(2)
- 46 to 47 labelled o_1(2)
- 46 to 64 labelled t_1(3)
- 46 to 68 labelled n_1(4)
- 46 to 88 labelled a_1(5)
- 46 to 91 labelled a_1(6)
- 47 to 48 labelled m_1(2)
- 50 to 21 labelled e_1(2)
- 50 to 34 labelled e_1(2)
- 49 to 50 labelled t_1(2)
- 49 to 56 labelled n_1(3)
- 49 to 69 labelled a_1(4)
- 49 to 86 labelled a_1(5)
- 45 to 46 labelled t_1(2)
- 45 to 55 labelled m_1(3)
- 48 to 49 labelled a_1(2)
- 44 to 45 labelled a_1(2)
- 51 to 52 labelled e_1(2)
- 52 to 30 labelled n_1(2)
- 53 to 42 labelled t_1(3)
- 54 to 39 labelled t_1(3)
- 55 to 47 labelled a_1(3)
- 56 to 21 labelled s_1(3)
- 56 to 34 labelled s_1(3)
- 57 to 52 labelled s_1(3)
- 60 to 2 labelled t_1(3)
- 60 to 24 labelled m_1(1)
- 60 to 18 labelled n_1(1)
- 60 to 38 labelled a_1(2)
- 60 to 54 labelled a_1(3)
- 58 to 59 labelled l_1(3)
- 59 to 60 labelled a_1(3)
- 63 to 33 labelled t_1(3)
- 61 to 62 labelled l_1(3)
- 62 to 63 labelled a_1(3)
- 64 to 65 labelled e_1(3)
- 65 to 49 labelled n_1(3)
- 66 to 59 labelled t_1(4)
- 67 to 62 labelled t_1(4)
- 68 to 65 labelled s_1(4)
- 71 to 21 labelled t_1(4)
- 71 to 34 labelled t_1(4)
- 70 to 71 labelled a_1(4)
- 69 to 70 labelled l_1(4)
- 74 to 52 labelled t_1(4)
- 73 to 74 labelled a_1(4)
- 72 to 73 labelled l_1(4)
- 86 to 70 labelled t_1(5)
- 87 to 73 labelled t_1(5)
- 90 to 65 labelled t_1(5)
- 88 to 89 labelled l_1(5)
- 89 to 90 labelled a_1(5)
- 91 to 89 labelled t_1(6)