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`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

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))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{g, fib} > sp > + > s
{g, fib} > np > pair
{g, fib} > np > + > s

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)
fib(x1) -> fib(x1)
sp(x1) -> sp(x1)
g(x1) -> g(x1)
pair(x1, x2) -> pair(x1, x2)
np(x1) -> np(x1)
+(x1, x2) -> +(x1, x2)

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

Dependency Pair:

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))

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:

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))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

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))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{g, fib} > sp > {+, np} > pair
{g, fib} > sp > {+, np} > s

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
s(x1) -> s(x1)
fib(x1) -> fib(x1)
sp(x1) -> sp(x1)
g(x1) -> g(x1)
pair(x1, x2) -> pair(x1, x2)
np(x1) -> np(x1)
+(x1, x2) -> +(x1, x2)

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

Dependency Pair:

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))

Using the Dependency Graph resulted in no new DP problems.

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