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(f(f(a))) → mark(f(g(f(a))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
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(f(f(a))) → mark(f(g(f(a))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
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 4. This implies Q-termination of R.
The following rules were used to construct the certificate:
active(f(f(a))) → mark(f(g(f(a))))
mark(f(X)) → active(f(mark(X)))
mark(a) → active(a)
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, 3, 4, 5, 6, 7, 9, 8, 11, 10, 12, 13, 14, 15, 17, 16, 18, 19, 21, 22, 20, 23, 32, 31, 33
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)
- 1 to 6 labelled active_1(0)
- 1 to 10 labelled active_1(1)
- 1 to 12 labelled mark_1(1), g_1(1), g_1(2), f_1(2), f_1(1)
- 1 to 7 labelled f_1(2), f_1(1), g_1(1), g_1(2)
- 1 to 8 labelled f_1(2), f_1(1), g_1(1), g_1(2)
- 1 to 16 labelled active_1(2), g_1(1), g_1(2), f_1(2), f_1(1)
- 1 to 20 labelled g_1(1), g_1(2), f_1(2), f_1(1)
- 1 to 31 labelled g_1(1), g_1(2), f_1(2), f_1(1)
- 2 to 2 labelled #_1(0), mark_1(0)
- 2 to 7 labelled active_1(1)
- 2 to 8 labelled active_1(1)
- 2 to 12 labelled mark_1(1)
- 2 to 16 labelled active_1(2)
- 2 to 20 labelled mark_1(2)
- 2 to 31 labelled active_1(3)
- 3 to 4 labelled f_1(0)
- 4 to 5 labelled g_1(0)
- 5 to 6 labelled f_1(0)
- 6 to 2 labelled a(0)
- 7 to 2 labelled g_1(1), a(1)
- 7 to 7 labelled g_1(2)
- 7 to 8 labelled g_1(2)
- 7 to 12 labelled g_1(2)
- 7 to 20 labelled g_1(2)
- 7 to 16 labelled g_1(2)
- 7 to 31 labelled g_1(2)
- 9 to 2 labelled mark_1(1)
- 9 to 1 labelled active_1(1)
- 9 to 8 labelled active_1(1)
- 9 to 7 labelled active_1(1)
- 9 to 12 labelled mark_1(1)
- 9 to 16 labelled active_1(2)
- 9 to 20 labelled mark_1(2)
- 9 to 31 labelled active_1(3)
- 8 to 9 labelled f_1(1)
- 8 to 2 labelled f_1(2), f_1(1)
- 8 to 8 labelled f_1(2)
- 8 to 7 labelled f_1(2)
- 8 to 1 labelled f_1(2), f_1(1)
- 8 to 3 labelled f_1(1)
- 8 to 12 labelled f_1(2)
- 8 to 16 labelled f_1(3), f_1(2)
- 8 to 10 labelled f_1(2)
- 8 to 6 labelled f_1(1)
- 8 to 20 labelled f_1(2), f_1(3)
- 8 to 31 labelled f_1(3), f_1(2)
- 11 to 4 labelled mark_1(1)
- 11 to 18 labelled active_1(1)
- 10 to 11 labelled f_1(1)
- 10 to 4 labelled f_1(2)
- 10 to 18 labelled f_1(2)
- 12 to 13 labelled f_1(1)
- 13 to 14 labelled g_1(1)
- 14 to 15 labelled f_1(1)
- 15 to 2 labelled a(1)
- 17 to 13 labelled mark_1(2)
- 17 to 19 labelled active_1(2)
- 16 to 17 labelled f_1(2)
- 16 to 13 labelled f_1(3)
- 16 to 19 labelled f_1(3)
- 18 to 5 labelled g_1(1)
- 19 to 14 labelled g_1(2)
- 21 to 22 labelled g_1(2)
- 22 to 23 labelled f_1(2)
- 20 to 21 labelled f_1(2)
- 23 to 2 labelled a(2)
- 32 to 21 labelled mark_1(3)
- 32 to 33 labelled active_1(3)
- 31 to 32 labelled f_1(3)
- 31 to 21 labelled f_1(4)
- 31 to 33 labelled f_1(4)
- 33 to 22 labelled g_1(3)