Term Rewriting System R:
[m, n, r]
p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

P(m, n, s(r)) -> P(m, r, n)
P(m, s(n), 0) -> P(0, n, m)

Furthermore, R contains one SCC.

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

Dependency Pairs:

P(m, s(n), 0) -> P(0, n, m)
P(m, n, s(r)) -> P(m, r, n)

Rules:

p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m

Strategy:

innermost

The following dependency pairs can be strictly oriented:

P(m, s(n), 0) -> P(0, n, m)
P(m, n, s(r)) -> P(m, r, n)

There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(P(x1, x2, x3)) =  x1 + x2 + x3 POL(0) =  0 POL(s(x1)) =  1 + x1

resulting in one new DP problem.

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

Dependency Pair:

Rules:

p(m, n, s(r)) -> p(m, r, n)
p(m, s(n), 0) -> p(0, n, m)
p(m, 0, 0) -> m

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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