Term Rewriting System R:
[y, x]
-(0, y) -> 0
-(x, 0) -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) -> 0
p(s(x)) -> x

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

-'(x, s(y)) -> -'(x, p(s(y)))
-'(x, s(y)) -> P(s(y))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pair:

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

Rules:

-(0, y) -> 0
-(x, 0) -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) -> 0
p(s(x)) -> x

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

-'(x, s(y)) -> -'(x, p(s(y)))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

-(0, y) -> 0
-(x, 0) -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) -> 0
p(s(x)) -> x

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-'(x1, x2)) =  x2 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

-(0, y) -> 0
-(x, 0) -> x
-(x, s(y)) -> if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) -> 0
p(s(x)) -> x

Using the Dependency Graph resulted in no new DP problems.

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