Term Rewriting System R:
[x, y]
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(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:

SUM(s(x)) -> +'(sum(x), s(x))
SUM(s(x)) -> SUM(x)
+'(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:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

• Dependency Pair:

SUM(s(x)) -> SUM(x)

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(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:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

• Dependency Pair:

SUM(s(x)) -> SUM(x)

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))

Termination of R could not be shown.
Duration:
0:00 minutes