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

Innermost 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
         ↳Narrowing Transformation


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


Strategy:

innermost




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

F(g(X), Y) -> F(X, nf(g(X), activate(Y)))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Non Termination


Dependency Pairs:

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


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


Strategy:

innermost




Found an infinite P-chain over R:
P =

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

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

Thus, s starts an infinite chain.

Innermost Termination of R could not be shown.
Duration:
0:06 minutes