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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

ACTIVATE(nf(nf(X1'', X2''), X2)) -> F(f(activate(X1''), X2''), X2)
ACTIVATE(nf(ng(X'), X2)) -> F(g(activate(X')), X2)
ACTIVATE(nf(X1', X2)) -> F(X1', X2)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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




Termination of R could not be shown.
Duration:
0:00 minutes