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(x1)) → d(x1)
b(a(x1)) → a(b(x1))
d(c(x1)) → f(a(b(b(c(x1)))))
d(f(x1)) → f(a(b(x1)))
a(f(x1)) → a(x1)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x1)) → d(x1)
b(a(x1)) → a(b(x1))
d(c(x1)) → f(a(b(b(c(x1)))))
d(f(x1)) → f(a(b(x1)))
a(f(x1)) → 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(x1)) → d(x1)
b(a(x1)) → a(b(x1))
d(c(x1)) → f(a(b(b(c(x1)))))
d(f(x1)) → f(a(b(x1)))
a(f(x1)) → a(x1)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 4, 3, 8, 7, 6, 5, 10, 9, 14, 13, 12, 11
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled f_1(0)
- 1 to 2 labelled d_1(0), a_1(0), d_1(1), a_1(1)
- 1 to 4 labelled a_1(0)
- 1 to 5 labelled f_1(0)
- 1 to 9 labelled f_1(1), a_1(1), a_1(2)
- 1 to 11 labelled f_1(1), a_1(1), a_1(2)
- 2 to 2 labelled #_1(0)
- 4 to 2 labelled b_1(0), d_1(2)
- 4 to 10 labelled a_1(1)
- 4 to 9 labelled f_1(1), a_1(2)
- 4 to 11 labelled f_1(1), a_1(2)
- 3 to 4 labelled a_1(0)
- 3 to 2 labelled d_1(1)
- 3 to 9 labelled f_1(1), a_1(1), a_1(2)
- 3 to 11 labelled f_1(1), a_1(1), a_1(2)
- 8 to 2 labelled c_1(0)
- 7 to 8 labelled b_1(0)
- 6 to 7 labelled b_1(0)
- 5 to 6 labelled a_1(0)
- 5 to 7 labelled d_1(1)
- 10 to 2 labelled b_1(1), d_1(2)
- 10 to 10 labelled a_1(1)
- 10 to 9 labelled f_1(1), a_1(2)
- 10 to 11 labelled f_1(1), a_1(2)
- 9 to 10 labelled a_1(1)
- 9 to 2 labelled d_1(2)
- 9 to 9 labelled f_1(1), a_1(2)
- 9 to 11 labelled f_1(1), a_1(2)
- 14 to 2 labelled c_1(1)
- 13 to 14 labelled b_1(1)
- 12 to 13 labelled b_1(1)
- 11 to 12 labelled a_1(1)
- 11 to 13 labelled d_1(2)