Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(X) -> G(X)
G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nf(X)) -> F(X)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

G(s(X)) -> G(X)

Rules:

f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(s(X)) -> G(X)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(G(x1)) =  x1 POL(s(x1)) =  1 + x1

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
s(x1) -> s(x1)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pair:

Rules:

f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

Rules:

f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X

Strategy:

innermost

The following dependency pair can be strictly oriented:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

activate(nf(X)) -> f(X)
activate(X) -> X
f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)

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

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

f(X) -> cons(X, nf(g(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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