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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(x, y), 0)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

Furthermore, R contains one SCC.

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

Dependency Pairs:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)
AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(x, y)

There are no usable rules w.r.t. the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
AP(x1, x2) -> x1
ap(x1, x2) -> ap(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Non Termination`

Dependency Pair:

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

Rules:

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

Strategy:

innermost

Found an infinite P-chain over R:
P =

AP(ap(ap(g, x), y), ap(s, z)) -> AP(ap(ap(g, x), y), ap(ap(x, y), 0))

R =

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

s = AP(ap(ap(g, f), s), ap(ap(f, s), 0))
evaluates to t =AP(ap(ap(g, f), s), ap(ap(f, s), 0))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:01 minutes