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:
b(c(a(x1))) → a(b(a(b(x1))))
b(x1) → c(c(x1))
a(a(x1)) → a(c(b(a(x1))))
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(c(a(x1))) → a(b(a(b(x1))))
b(x1) → c(c(x1))
a(a(x1)) → a(c(b(a(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:
b(c(a(x1))) → a(b(a(b(x1))))
b(x1) → c(c(x1))
a(a(x1)) → a(c(b(a(x1))))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 9, 7, 8, 10, 11, 12, 15, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29
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 6 labelled c_1(0)
- 1 to 7 labelled a_1(0)
- 2 to 2 labelled #_1(0)
- 3 to 4 labelled c_1(0)
- 4 to 5 labelled b_1(0)
- 4 to 12 labelled c_1(1)
- 5 to 2 labelled a_1(0)
- 5 to 16 labelled a_1(1)
- 6 to 2 labelled c_1(0)
- 9 to 2 labelled b_1(0)
- 9 to 11 labelled c_1(1)
- 9 to 13 labelled a_1(1)
- 7 to 8 labelled b_1(0)
- 7 to 10 labelled c_1(1)
- 8 to 9 labelled a_1(0)
- 8 to 22 labelled a_1(1)
- 10 to 8 labelled c_1(1)
- 11 to 2 labelled c_1(1)
- 12 to 5 labelled c_1(1)
- 15 to 2 labelled b_1(1)
- 15 to 19 labelled c_1(2)
- 15 to 13 labelled a_1(1)
- 13 to 14 labelled b_1(1)
- 13 to 20 labelled c_1(2)
- 14 to 15 labelled a_1(1)
- 14 to 26 labelled a_1(2)
- 16 to 17 labelled c_1(1)
- 17 to 18 labelled b_1(1)
- 17 to 21 labelled c_1(2)
- 18 to 2 labelled a_1(1)
- 18 to 16 labelled a_1(1)
- 19 to 2 labelled c_1(2)
- 20 to 14 labelled c_1(2)
- 21 to 18 labelled c_1(2)
- 22 to 23 labelled c_1(1)
- 23 to 24 labelled b_1(1)
- 23 to 25 labelled c_1(2)
- 24 to 13 labelled a_1(1)
- 25 to 24 labelled c_1(2)
- 26 to 27 labelled c_1(2)
- 27 to 28 labelled b_1(2)
- 27 to 29 labelled c_1(3)
- 28 to 13 labelled a_1(2)
- 29 to 28 labelled c_1(3)