Term Rewriting System R:

   [X]
   active(g(X)) -> mark(h(X))
   active(c) -> mark(d)
   active(h(d)) -> mark(g(c))
   proper(g(X)) -> g(proper(X))
   proper(h(X)) -> h(proper(X))
   proper(c) -> ok(c)
   proper(d) -> ok(d)
   g(ok(X)) -> ok(g(X))
   h(ok(X)) -> ok(h(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 4.
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, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26

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 2 labelled h(0)
* 1 to 4 labelled mark(0)
* 4 to 2 labelled d(0)
* 1 to 5 labelled mark(0)
* 5 to 6 labelled g(0)
* 6 to 2 labelled c(0)
* 1 to 7 labelled g(0)
* 7 to 2 labelled proper(0)
* 1 to 7 labelled h(0)
* 1 to 6 labelled ok(0)
* 1 to 4 labelled ok(0)
* 1 to 8 labelled ok(0)
* 8 to 2 labelled g(0)
* 1 to 3 labelled ok(0)
* 1 to 7 labelled top(0)
* 1 to 9 labelled top(0)
* 9 to 2 labelled active(0)
* 9 to 10 labelled mark(1)
* 10 to 2 labelled h(1)
* 9 to 11 labelled mark(1)
* 11 to 12 labelled g(1)
* 12 to 2 labelled c(1)
* 8 to 13 labelled ok(1)
* 13 to 2 labelled g(1)
* 7 to 14 labelled h(1)
* 14 to 2 labelled proper(1)
* 7 to 14 labelled g(1)
* 3 to 10 labelled ok(1)
* 1 to 15 labelled top(1)
* 15 to 11 labelled proper(1)
* 10 to 10 labelled ok(1)
* 13 to 13 labelled ok(1)
* 14 to 14 labelled h(1)
* 14 to 14 labelled g(1)
* 15 to 10 labelled proper(1)
* 15 to 16 labelled h(2)
* 16 to 2 labelled proper(2)
* 15 to 17 labelled g(2)
* 17 to 12 labelled proper(2)
* 16 to 14 labelled h(1)
* 16 to 14 labelled g(1)
* 17 to 18 labelled ok(2)
* 18 to 2 labelled c(2)
* 15 to 19 labelled ok(3)
* 19 to 18 labelled g(3)
* 1 to 20 labelled top(2)
* 20 to 19 labelled active(2)
* 20 to 21 labelled mark(3)
* 21 to 18 labelled h(3)
* 1 to 22 labelled top(3)
* 22 to 21 labelled proper(3)
* 22 to 23 labelled h(4)
* 23 to 18 labelled proper(4)
* 23 to 24 labelled ok(3)
* 24 to 2 labelled c(3)
* 22 to 25 labelled ok(4)
* 25 to 24 labelled h(4)
* 1 to 26 labelled top(4)
* 26 to 25 labelled active(4)




Termination of R successfully shown.


Duration: 0.453 seconds.