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