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(b(a(a(x1)))) → c(b(a(b(a(x1)))))
a(c(b(x1))) → a(a(b(c(b(a(x1))))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(a(a(x1)))) → c(b(a(b(a(x1)))))
a(c(b(x1))) → a(a(b(c(b(a(x1))))))
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:
a(b(a(a(x1)))) → c(b(a(b(a(x1)))))
a(c(b(x1))) → a(a(b(c(b(a(x1))))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 6, 7, 4, 3, 8, 10, 11, 9, 14, 15, 16, 13, 12, 17, 19, 20, 18, 21, 23, 24, 22
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled a_1(0)
- 1 to 8 labelled c_1(0)
- 2 to 2 labelled #_1(0)
- 5 to 6 labelled c_1(0)
- 6 to 7 labelled b_1(0)
- 7 to 2 labelled a_1(0)
- 7 to 12 labelled a_1(1)
- 7 to 17 labelled c_1(1)
- 4 to 5 labelled b_1(0)
- 3 to 4 labelled a_1(0)
- 8 to 9 labelled b_1(0)
- 10 to 11 labelled b_1(0)
- 11 to 2 labelled a_1(0)
- 11 to 12 labelled a_1(1)
- 11 to 17 labelled c_1(1)
- 9 to 10 labelled a_1(0)
- 9 to 17 labelled c_1(1)
- 14 to 15 labelled c_1(1)
- 15 to 16 labelled b_1(1)
- 16 to 2 labelled a_1(1)
- 16 to 12 labelled a_1(1)
- 16 to 17 labelled c_1(1)
- 13 to 14 labelled b_1(1)
- 12 to 13 labelled a_1(1)
- 17 to 18 labelled b_1(1)
- 19 to 20 labelled b_1(1)
- 20 to 2 labelled a_1(1)
- 20 to 12 labelled a_1(1)
- 20 to 17 labelled c_1(1)
- 20 to 13 labelled a_1(1)
- 18 to 19 labelled a_1(1)
- 18 to 17 labelled c_1(1)
- 18 to 21 labelled c_1(2)
- 21 to 22 labelled b_1(2)
- 23 to 24 labelled b_1(2)
- 24 to 13 labelled a_1(2)
- 22 to 23 labelled a_1(2)