Term Rewriting System R:
[x, y, z]
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`
`         ↳Size-Change Principle`

Dependency Pairs:

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

Rules:

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

We number the DPs as follows:
1. F(x, y, s(z)) -> F(0, 1, z)
2. F(0, 1, x) -> F(s(x), x, x)
and get the following Size-Change Graph(s):
{1} , {1}
3>3
{2} , {2}
3=2
3=3

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
3>3
{2} , {1}
3>3
{1} , {2}
3>2
3>3

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
s(x1) -> s(x1)

We obtain no new DP problems.

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