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__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)
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__f(f(a)) → c(f(g(f(a))))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(c(X)) → c(X)
mark(g(X)) → g(mark(X))
a__f(X) → f(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 2 labelled c_1(0), a(0), f_1(0)
- 1 to 3 labelled a__f_1(0), g_1(0), f_1(1)
- 1 to 4 labelled c_1(0)
- 1 to 9 labelled c_1(1)
- 2 to 2 labelled #_1(0)
- 3 to 2 labelled mark_1(0), c_1(1), a(1)
- 3 to 8 labelled a__f_1(1), g_1(1), f_1(2)
- 3 to 13 labelled c_1(2)
- 4 to 5 labelled f_1(0)
- 5 to 6 labelled g_1(0)
- 6 to 7 labelled f_1(0)
- 7 to 2 labelled a(0)
- 8 to 2 labelled mark_1(1), c_1(1), a(1)
- 8 to 8 labelled a__f_1(1), g_1(1), f_1(2)
- 8 to 13 labelled c_1(2)
- 9 to 10 labelled f_1(1)
- 10 to 11 labelled g_1(1)
- 11 to 12 labelled f_1(1)
- 12 to 2 labelled a(1)
- 13 to 14 labelled f_1(2)
- 14 to 15 labelled g_1(2)
- 15 to 16 labelled f_1(2)
- 16 to 2 labelled a(2)