Term Rewriting System R:
[X]
f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ACTIVATE(nh(X)) -> H(activate(X))
ACTIVATE(nh(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nh(X)) -> ACTIVATE(X)

Rules:

f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X

The following dependency pairs can be strictly oriented:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nh(X)) -> ACTIVATE(X)

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nh(x1) -> nh(x1)
nf(x1) -> nf(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes