Term Rewriting System R:
[X]
f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(s(0)) -> F(p(s(0)))
F(s(0)) -> P(s(0))
ACTIVATE(nf(X)) -> F(X)

Furthermore, R contains one SCC.

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

Dependency Pair:

F(s(0)) -> F(p(s(0)))

Rules:

f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X

The following dependency pair can be strictly oriented:

F(s(0)) -> F(p(s(0)))

The following rules can be oriented:

p(s(0)) -> 0
f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
activate(nf(X)) -> f(X)
activate(X) -> X

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(n__f(x1)) =  x1 POL(activate(x1)) =  1 + x1 POL(0) =  0 POL(cons(x1, x2)) =  x1 + x2 POL(s(x1)) =  1 + x1 POL(F(x1)) =  x1 POL(f(x1)) =  1 + x1 POL(p) =  0

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
s(x1) -> s(x1)
p(x1) -> p
f(x1) -> f(x1)
cons(x1, x2) -> cons(x1, x2)
nf(x1) -> nf(x1)
activate(x1) -> activate(x1)

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

Dependency Pair:

Rules:

f(0) -> cons(0, nf(s(0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
activate(nf(X)) -> f(X)
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

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