Term Rewriting System R:
[X, Y, X1, X2]
f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Non Termination


Dependency Pairs:

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


Rules:


f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X





Found an infinite P-chain over R:
P =

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

R =

f(g(X), Y) -> f(X, nf(g(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
activate(nf(X1, X2)) -> f(X1, X2)
activate(X) -> X

s = F(g(g(X'')), Y')
evaluates to t =F(g(g(X'')), activate(Y'))

Thus, s starts an infinite chain as s matches t.

Non-Termination of R could be shown.
Duration:
0:00 minutes