Term Rewriting System R:
[x, y, z]
g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(0, 1, x) -> F(s(x), x, x)
F(x, y, s(z)) -> F(0, 1, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(x, y, s(z)) -> F(0, 1, z)
F(0, 1, x) -> F(s(x), x, x)

Rules:

g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

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

F(x, y, s(z)) -> F(0, 1, z)
two new Dependency Pairs are created:

F(0, 1, s(z'')) -> F(0, 1, z'')
F(s(x'''), y', s(z')) -> F(0, 1, z')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

F(0, 1, s(z'')) -> F(0, 1, z'')
F(s(x'''), y', s(z')) -> F(0, 1, z')
F(0, 1, x) -> F(s(x), x, x)

Rules:

g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

The following dependency pairs can be strictly oriented:

F(0, 1, s(z'')) -> F(0, 1, z'')
F(s(x'''), y', s(z')) -> F(0, 1, z')

The following rules can be oriented:

g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> x3
s(x1) -> s(x1)
g(x1, x2) -> g(x1, x2)
f(x1, x2, x3) -> x3

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

Dependency Pair:

F(0, 1, x) -> F(s(x), x, x)

Rules:

g(x, y) -> x
g(x, y) -> y
f(0, 1, x) -> f(s(x), x, x)
f(x, y, s(z)) -> s(f(0, 1, z))

Using the Dependency Graph resulted in no new DP problems.

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