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:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(x1))) → a(l(c(c(r(x1)))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(x1))) → a(l(c(c(r(x1)))))
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(x1))) → a(l(c(c(r(x1)))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 6, 7, 5, 4, 8, 9, 10, 11, 14, 15, 13, 12, 16, 19, 20, 18, 17, 21, 22, 23, 24, 25, 26
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), r_1(0), c_1(0)
- 1 to 4 labelled a_1(0)
- 1 to 9 labelled l_1(1)
- 1 to 8 labelled r_1(1)
- 1 to 12 labelled a_1(1)
- 1 to 16 labelled l_1(2)
- 2 to 2 labelled #_1(0)
- 3 to 2 labelled a_1(0), a_1(2)
- 3 to 8 labelled l_1(1), c_1(1), r_1(1)
- 3 to 17 labelled a_1(2)
- 3 to 21 labelled l_1(3)
- 6 to 7 labelled c_1(0)
- 7 to 2 labelled r_1(0)
- 5 to 6 labelled c_1(0)
- 4 to 5 labelled l_1(0)
- 8 to 2 labelled a_1(1)
- 8 to 8 labelled l_1(1), c_1(1), r_1(1)
- 8 to 17 labelled a_1(2)
- 8 to 21 labelled l_1(3)
- 9 to 5 labelled a_1(1)
- 9 to 10 labelled c_1(1)
- 10 to 6 labelled a_1(1)
- 10 to 11 labelled c_1(1)
- 10 to 8 labelled r_1(1)
- 11 to 7 labelled a_1(1)
- 14 to 15 labelled c_1(1)
- 15 to 2 labelled r_1(1)
- 15 to 17 labelled r_1(1)
- 13 to 14 labelled c_1(1)
- 12 to 13 labelled l_1(1)
- 16 to 13 labelled a_1(2)
- 16 to 22 labelled c_1(2)
- 19 to 20 labelled c_1(2)
- 20 to 2 labelled r_1(2)
- 20 to 17 labelled r_1(2)
- 18 to 19 labelled c_1(2)
- 17 to 18 labelled l_1(2)
- 21 to 18 labelled a_1(3)
- 21 to 23 labelled c_1(3)
- 22 to 14 labelled a_1(2)
- 22 to 24 labelled c_1(2)
- 22 to 3 labelled r_1(2)
- 23 to 19 labelled a_1(3)
- 23 to 25 labelled c_1(3)
- 23 to 26 labelled r_1(3)
- 24 to 15 labelled a_1(2)
- 25 to 20 labelled a_1(3)
- 26 to 17 labelled a_1(3), a_1(2)
- 26 to 2 labelled a_1(3)
- 26 to 21 labelled l_1(3)
- 26 to 8 labelled l_1(1), c_1(1), r_1(1)