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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

The following dependency pair can be strictly oriented:

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

There are no usable rules using the Ce-refinement that need to be oriented.

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

resulting in one new DP problem.

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

Dependency Pair:

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

Using the Dependency Graph resulted in no new DP problems.

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

The following remains to be proven:
Dependency Pairs:

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

Rules:

-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
p(s(x)) -> x
f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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