Term Rewriting System R:

   [X]
   active(c) -> mark(f(g(c)))
   active(f(g(X))) -> mark(g(X))
   proper(c) -> ok(c)
   proper(f(X)) -> f(proper(X))
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Termination of R to be shown.


Termination could be shown with a Match Bound of 2.
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

Node 1 is start node and node 2 is final node.

Those nodes are connect through the following edges:

* 2 to 2 labelled #(0)
* 1 to 3 labelled mark(0)
* 3 to 4 labelled f(0)
* 4 to 5 labelled g(0)
* 5 to 2 labelled c(0)
* 1 to 6 labelled mark(0)
* 6 to 2 labelled g(0)
* 1 to 5 labelled ok(0)
* 1 to 7 labelled f(0)
* 7 to 2 labelled proper(0)
* 1 to 7 labelled g(0)
* 1 to 8 labelled ok(0)
* 8 to 2 labelled f(0)
* 1 to 6 labelled ok(0)
* 1 to 7 labelled top(0)
* 1 to 9 labelled top(0)
* 9 to 2 labelled active(0)
* 8 to 10 labelled ok(1)
* 10 to 2 labelled f(1)
* 7 to 11 labelled f(1)
* 11 to 2 labelled proper(1)
* 7 to 11 labelled g(1)
* 9 to 12 labelled mark(1)
* 12 to 2 labelled g(1)
* 6 to 12 labelled ok(1)
* 10 to 10 labelled ok(1)
* 11 to 11 labelled f(1)
* 11 to 11 labelled g(1)
* 12 to 12 labelled ok(1)
* 1 to 13 labelled top(1)
* 13 to 12 labelled proper(1)
* 13 to 14 labelled g(2)
* 14 to 2 labelled proper(2)
* 14 to 11 labelled f(1)
* 14 to 11 labelled g(1)




Termination of R successfully shown.


Duration: 0.372 seconds.