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:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
mark(c) → active(c)
mark(f(X)) → active(f(X))
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
mark(c) → active(c)
mark(f(X)) → active(f(X))
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
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:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
mark(c) → active(c)
mark(f(X)) → active(f(X))
mark(g(X)) → active(g(X))
f(mark(X)) → f(X)
f(active(X)) → f(X)
g(mark(X)) → g(X)
g(active(X)) → g(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 5, 3, 4, 8, 6, 7, 9, 10, 11
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 2 labelled g_1(0), f_1(0), g_1(1), f_1(1)
- 1 to 3 labelled mark_1(0)
- 1 to 1 labelled active_1(0), mark_1(0), active_1(1), mark_1(1)
- 1 to 5 labelled active_1(0), g_1(1)
- 1 to 6 labelled mark_1(1)
- 1 to 4 labelled f_1(1)
- 1 to 9 labelled active_1(2)
- 1 to 10 labelled mark_1(2)
- 1 to 11 labelled active_1(3)
- 2 to 2 labelled #_1(0)
- 5 to 2 labelled c(0)
- 3 to 4 labelled f_1(0)
- 4 to 5 labelled g_1(0)
- 8 to 2 labelled c(1)
- 6 to 7 labelled f_1(1)
- 7 to 8 labelled g_1(1)
- 9 to 2 labelled g_1(2), f_1(2), f_1(1), g_1(1)
- 9 to 7 labelled f_1(2)
- 9 to 4 labelled f_1(2)
- 9 to 5 labelled g_1(2)
- 10 to 8 labelled g_1(2)
- 11 to 8 labelled g_1(3)