Term Rewriting System R:

   [X, Y, X1, X2]
   f(g(X), Y) -> f(X, n__f(n__g(X), activate(Y)))
   f(X1, X2) -> n__f(X1, X2)
   g(X) -> n__g(X)
   activate(n__f(X1, X2)) -> f(activate(X1), X2)
   activate(n__g(X)) -> g(activate(X))
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__f(X1, X2)) -> F(activate(X1), X2)
   ACTIVATE(n__f(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__g(X)) -> G(activate(X))
   ACTIVATE(n__g(X)) -> ACTIVATE(X)
   F(g(X), Y) -> F(X, n__f(n__g(X), activate(Y)))
   F(g(X), Y) -> ACTIVATE(Y)

Furthermore, R contains one SCC.


SCC1:

ACTIVATE(n__g(X)) -> ACTIVATE(X)
ACTIVATE(n__f(X1, X2)) -> ACTIVATE(X1)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, n__f(n__g(X), activate(Y)))
ACTIVATE(n__f(X1, X2)) -> F(activate(X1), X2)

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   ACTIVATE(n__f(X1, X2)) -> F(activate(X1), X2)
three new Dependency Pairs
are created:


   ACTIVATE(n__f(n__f(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
   ACTIVATE(n__f(n__g(X'), X2)) -> F(g(activate(X')), X2)
   ACTIVATE(n__f(X1', X2)) -> F(X1', X2)

The transformation is resulting in one subcycle:


SCC1.Nar1:

ACTIVATE(n__f(X1', X2)) -> F(X1', X2)
ACTIVATE(n__f(n__g(X'), X2)) -> F(g(activate(X')), X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, n__f(n__g(X), activate(Y)))
ACTIVATE(n__f(n__f(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
ACTIVATE(n__f(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n__g(X)) -> ACTIVATE(X)

Found an infinite P-chain over R:
P = ACTIVATE(n__f(X1', X2)) -> F(X1', X2)
ACTIVATE(n__f(n__g(X'), X2)) -> F(g(activate(X')), X2)
F(g(X), Y) -> ACTIVATE(Y)
F(g(X), Y) -> F(X, n__f(n__g(X), activate(Y)))
ACTIVATE(n__f(n__f(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
ACTIVATE(n__f(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n__g(X)) -> ACTIVATE(X)
R = [activate(n__f(X1, X2)) -> f(activate(X1), X2), activate(n__g(X)) -> g(activate(X)), activate(X) -> X, f(g(X), Y) -> f(X, n__f(n__g(X), activate(Y))), f(X1, X2) -> n__f(X1, X2), g(X) -> n__g(X)]
s = ACTIVATE(n__f(n__g(g(X')), X2'')) evaluates to t = ACTIVATE(n__f(n__g(g(X')), activate(X2'')))
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 15.73 seconds.