Term Rewriting System R:
[x]
sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

SUM(s(x)) -> SUM(x)
SUM1(s(x)) -> SUM1(x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

The following dependency pair can be strictly oriented:

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

The following rules can be oriented:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
sum > +
sum1 > +
sum1 > s

resulting in one new DP problem.
Used Argument Filtering System:
SUM(x1) -> SUM(x1)
s(x1) -> s(x1)
sum(x1) -> sum(x1)
+(x1, x2) -> +(x1, x2)
sum1(x1) -> sum1(x1)

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

Dependency Pair:

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(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:

SUM1(s(x)) -> SUM1(x)

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

The following dependency pair can be strictly oriented:

SUM1(s(x)) -> SUM1(x)

The following rules can be oriented:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
sum > +
sum1 > +
sum1 > s

resulting in one new DP problem.
Used Argument Filtering System:
SUM1(x1) -> SUM1(x1)
s(x1) -> s(x1)
sum(x1) -> sum(x1)
+(x1, x2) -> +(x1, x2)
sum1(x1) -> sum1(x1)

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

Dependency Pair:

Rules:

sum(0) -> 0
sum(s(x)) -> +(sum(x), s(x))
sum1(0) -> 0
sum1(s(x)) -> s(+(sum1(x), +(x, x)))

Using the Dependency Graph resulted in no new DP problems.

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