Term Rewriting System R:
[N, X, Y, X1, X2, XS]
fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FIB(N) -> SEL(N, fib1(s(0), s(0)))
FIB(N) -> FIB1(s(0), s(0))
FIB1(X, Y) -> ADD(X, Y)
SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))
SEL(s(N), cons(X, XS)) -> ACTIVATE(XS)
ACTIVATE(nfib1(X1, X2)) -> FIB1(X1, X2)

Furthermore, R contains two SCCs.

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

Dependency Pair:

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

The following rules can be oriented:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
fib > sel > activate > fib1 > add > s > 0
fib > sel > activate > fib1 > cons > 0
fib > sel > activate > fib1 > nfib1 > 0

resulting in one new DP problem.
Used Argument Filtering System:
s(x1) -> s(x1)
fib(x1) -> fib(x1)
sel(x1, x2) -> sel(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
cons(x1, x2) -> cons(x1, x2)
nfib1(x1, x2) -> nfib1(x1, x2)
activate(x1) -> activate(x1)

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

Dependency Pair:

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

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(N), cons(X, XS)) -> SEL(N, activate(XS))

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

The following dependency pair can be strictly oriented:

SEL(s(N), cons(X, XS)) -> SEL(N, activate(XS))

The following rules can be oriented:

activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
fib(N) -> sel(N, fib1(s(0), s(0)))
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
SEL > {activate, fib1} > add > s
SEL > {activate, fib1} > cons
SEL > {activate, fib1} > nfib1
{0, fib} > sel > {activate, fib1} > add > s
{0, fib} > sel > {activate, fib1} > cons
{0, fib} > sel > {activate, fib1} > nfib1

resulting in one new DP problem.
Used Argument Filtering System:
SEL(x1, x2) -> SEL(x1, x2)
s(x1) -> s(x1)
cons(x1, x2) -> cons(x1, x2)
activate(x1) -> activate(x1)
nfib1(x1, x2) -> nfib1(x1, x2)
fib1(x1, x2) -> fib1(x1, x2)
fib(x1) -> fib(x1)
sel(x1, x2) -> sel(x1, x2)

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

Dependency Pair:

Rules:

fib(N) -> sel(N, fib1(s(0), s(0)))
fib1(X, Y) -> cons(X, nfib1(Y, add(X, Y)))
fib1(X1, X2) -> nfib1(X1, X2)
add(0, X) -> X
sel(0, cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
activate(nfib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Using the Dependency Graph resulted in no new DP problems.

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