Term Rewriting System R:
[x, y]
fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

FIB(s(s(x))) -> SP(g(x))
FIB(s(s(x))) -> G(x)
G(s(x)) -> NP(g(x))
G(s(x)) -> G(x)
SP(pair(x, y)) -> +'(x, y)
NP(pair(x, y)) -> +'(x, y)
+'(x, s(y)) -> +'(x, y)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

+'(x, s(y)) -> +'(x, y)

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

• Dependency Pair:

G(s(x)) -> G(x)

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Remaining Obligation(s)`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
• Dependency Pair:

+'(x, s(y)) -> +'(x, y)

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

• Dependency Pair:

G(s(x)) -> G(x)

Rules:

fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x))) -> sp(g(x))
g(0) -> pair(s(0), 0)
g(s(0)) -> pair(s(0), s(0))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +(x, y)
np(pair(x, y)) -> pair(+(x, y), x)
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Termination of R could not be shown.
Duration:
0:06 minutes